Addition and subtraction of degrees
Obviously, numbers with degrees can be accurate as other values , by adding them one after another with their signs.
So, the sum A 3 and B 2 is a 3 + b 2.
The sum A 3 - B n and H 5 -D 4 is A 3 - B N + H 5 - D 4.
Factors identical degrees of the same variables May be designed or deducted.
Thus, the amount 2a 2 and 3a 2 is 5a 2.
It is also obvious that if you take two squares a, or three squares a, or five squares a.
But degrees different variables and various degrees identical variablesmust be made by their addition with their signs.
So, the sum A 2 and A 3 is the sum A 2 + A 3.
This is obvious that the square of the number A, and the Cube of the number A, not equal to a double square A, but a double Cuba a.
Amount A 3 B n and 3a 5 B 6 is A 3 B N + 3A 5 B 6.
Subtraction The degrees are carried out in the same way as addition, except that the signs of subtractable must be changed accordingly.
Or:
2a 4 - (-6a 4) \u003d 8A 4
3H 2 B 6 - 4H 2 B 6 \u003d -H 2 B 6
5 (a - h) 6 - 2 (a - h) 6 \u003d 3 (a - h) 6
Multiplying degrees
Numbers with degrees can be multiplied by other values \u200b\u200bby writing them one after another, with a sign of multiplication or without it between them.
Thus, the result of multiplication A 3 on B 2 is a 3 b 2 or aaabb.
Or:
x -3 ⋅ a m \u003d a m x -3
3A 6 Y 2 ⋅ (-2x) \u003d -6A 6 XY 2
a 2 B 3 Y 2 ⋅ A 3 B 2 Y \u003d A 2 B 3 Y 2 A 3 B 2 y
The result in the latter example can be ordered by the addition of the same variables.
The expression will take the form: A 5 B 5 Y 3.
Comparing several numbers (variables) with degrees, we can see that if any two of them are multiplied, the result is the number (variable) with a degree equal to sum Degrees of the terms.
So, a 2 .a 3 \u003d aa.aaa \u003d aaaaa \u003d a 5.
Here 5 is the degree of multiplication result, equal to 2 + 3, the sum of the degrees of the components.
So, a n .a m \u003d a m + n.
For a n, A is taken as the multiplier as many times as the degree n;
And a m, takes as a multiplier as many times as the degree m;
Therefore, the degrees with the same bases can be multiplied by the addition of degrees.
So, a 2 .a 6 \u003d a 2 + 6 \u003d a 8. And x 3 .x 2 .x \u003d x 3 + 2 + 1 \u003d x 6.
Or:
4A N ⋅ 2a n \u003d 8a 2n
b 2 y 3 ⋅ b 4 y \u003d b 6 y 4
(B + H - Y) N ⋅ (B + H - Y) \u003d (B + H - Y) n + 1
Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).
This rule is valid for numbers, the degree of which - negative.
1. So, A -2 .a -3 \u003d a -5. This can be written in the form (1 / AA). (1 / AAA) \u003d 1 / AAAAA.
2. Y -n .y -m \u003d y -n-m.
3. A -N .A M \u003d A M-N.
If A + B is multiplied by A - B, the result will be equal to A 2 - B 2: That is
The result of multiplication of the amount or difference of two numbers is equal to the sum or difference of their squares.
If the sum is multiplied and the difference of two numbers erected into square, the result will be equal to the amount or difference of these numbers in fourth degree.
So, (a - y). (A + Y) \u003d A 2 - Y 2.
(A 2 - Y 2) ⋅ (A 2 + Y 2) \u003d A 4 \u200b\u200b- Y 4.
(A 4 - Y 4) ⋅ (a 4 + y 4) \u003d a 8 - y 8.
Decision degrees
The numbers with degrees can be divided, like other numbers, by taking a divide divider, or the placement of them in the form of a fraction.
Thus, a 3 b 2 divided by b 2, equal to A 3.
Recording A 5 divided by A 3 looks like $ \\ FRAC $. But this is equal to A 2. In a number of numbers
a +4, A +3, A +2, A +1, A 0, A -1, A -2, A -3, A -4.
any number can be divided into another, and the degree will be equal to difference Indicators of divisible numbers.
When dividing degrees with the same base, their indicators are deducted..
So, y 3: y 2 \u003d y 3-2 \u003d y 1. That is, $ \\ frac \u003d y $.
And a n + 1: a \u003d a n + 1-1 \u003d a n. That is, $ \\ frac \u003d a ^ n $.
Or:
y 2m: y m \u003d y m
8A n + m: 4a m \u003d 2a n
12 (B + Y) N: 3 (B + Y) 3 \u003d 4 (B + Y) N-3
The rule is also fair and for numbers with negative values \u200b\u200bof degrees.
The result of division A -5 on A -3 is equal to A -2.
Also, $ \\ FRAC: \\ FRAC \u003d \\ FRAC. \\ FRAC \u003d \\ FRAC \u003d \\ FRAC $.
h 2: H -1 \u003d H 2 + 1 \u003d H 3 or $ H ^ 2: \\ FRAC \u003d H ^ 2. \\ FRAC \u003d H ^ $ 3
It is necessary to very well assimilate the multiplication and division of degrees, as such operations are very widely used in algebra.
Examples of solving examples with fractions containing numbers with degrees
1. Reduce degrees in $ \\ FRAC $ reply: $ \\ FRAC $.
2. Reduce degrees in $ \\ FRAC $. Answer: $ \\ FRAC $ or 2x.
3. Reduce the degrees of A 2 / A 3 and A -3 / A -4 and bring to a common denominator.
a 2 .A -4 is a -2 first numerator.
a 3 .A -3 is a 0 \u003d 1, the second numerator.
a 3 .A -4 is a -1, a common numerator.
After simplification: A -2 / A -1 and 1 / A -1.
4. Reduce the indicators of the degrees 2a 4 / 5a 3 and 2 / a 4 and bring to the common denominator.
Answer: 2A 3 / 5A 7 and 5A 5 / 5A 7 or 2A 3 / 5A 2 and 5 / 5A 2.
5. Multiply (A 3 + B) / B 4 on (A - B) / 3.
6. Multiply (A 5 + 1) / x 2 on (B 2 - 1) / (X + A).
7. Multiply B 4 / A -2 on H -3 / X and A N / Y -3.
8. Divide A 4 / Y 3 on A 3 / Y 2. Answer: A / Y.
Properties of degree
We remind you that in this lesson you understand properties of degrees with natural indicators and zero. The degrees with rational indicators and their properties will be considered in lessons for 8 classes.
The ratio with a natural indicator has several important properties that allow you to simplify calculations in examples with degrees.
Property number 1.
The work of degrees
When multiplying degrees with the same bases, the base remains unchanged, and the indicators of degrees are folded.
a m · a n \u003d a m + n, where "A" is any number, and "M", "N" - any natural numbers.
This property of degrees also acts on the work of three and more degrees.
- Simplify the expression.
b · b 2 · b 3 · b 4 · b 5 \u003d b 1 + 2 + 3 + 4 + 5 \u003d b 15 - Represent in the form of degree.
6 15 · 36 \u003d 6 15 · 6 2 \u003d 6 15 · 6 2 \u003d 6 17 - Represent in the form of degree.
(0.8) 3 · (0.8) 12 \u003d (0.8) 3 + 12 \u003d (0.8) 15 - Write private in the form of degree
(2b) 5: (2b) 3 \u003d (2b) 5 - 3 \u003d (2b) 2 - Calculate.
Note that in the specified property it was only about multiplying degrees with the same bases. . It does not apply to their addition.
It is impossible to replace the amount (3 3 + 3 2) by 3 5. This is understandable if
calculate (3 3 + 3 2) \u003d (27 + 9) \u003d 36, a 3 5 \u003d 243
Property number 2.
Private degree
When dividing degrees with the same bases, the base remains unchanged, and from the indicator of the division deductible the degree of divider.
11 3 - 2 · 4 2 - 1 \u003d 11 · 4 \u003d 44
Example. Solve equation. We use the property of private degrees.
3 8: T \u003d 3 4
Answer: T \u003d 3 4 \u003d 81
Using properties No. 1 and No. 2, you can easily simplify expressions and make calculations.
Example. Simplify the expression.
4 5m + 6 · 4 m + 2: 4 4m + 3 \u003d 4 5m + 6 + m + 2: 4 4m + 3 \u003d 4 6m + 8 - 4m - 3 \u003d 4 2m + 5
Example. Find the value of the expression using the degree properties.
2 11 − 5 = 2 6 = 64
Please note that in the property 2 it was only about dividing degrees with the same bases.
It is impossible to replace the difference (4 3 -4 2) by 4 1. This is understandable if you calculate (4 3 -4 2) \u003d (64 - 16) \u003d 48, a 4 1 \u003d 4
Property number 3.
Erect
When erecting the degree to the degree, the foundation remains unchanged, and the indicators of degrees are variable.
(a n) m \u003d a n · m, where "A" is any number, and "M", "N" - any natural numbers.
We remind you that private can be represented as a fraction. Therefore, on the topic, we will focus more in more detail on the next page.
How to multiply degrees
How to multiply the degree? What degrees can multiply, and which is not? How to multiply the degree?
In the algebra to find a product of degrees in two cases:
1) if the degrees have the same bases;
2) If the degrees have the same indicators.
When multiplying degrees with the same bases, it is necessary to leave the base for the same, and the indicators are folded:
When multiplying degrees with the same indicators, the overall indicator can be reached by braces:
Consider how to multiply the degrees on specific examples.
The unit is not written in an indicator, but when multiplying degrees - take into account:
When multiplying, the number of degrees can be any. It should be remembered that in front of the lettering sign of multiplication can not write:
In expressions, the construction of the extent is performed first.
If the number is needed to multiply to the degree, you must first be raised to the degree, and only later - multiplication:
Multiplication of degrees with the same bases
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In this lesson, we will study the multiplication of degrees with the same bases. First, let us remember the degree of degree and formulate the theorem of equity equity . Then we give examples of its use on specific numbers and prove it. We also apply the theorem to solve various tasks.
Topic: degree with a natural indicator and its properties
Lesson: multiplication of degrees with the same bases (formula)
1. Basic definitions
Main definitions:
n. - indicator,
— n.The degree of date.
2. The wording of Theorem 1
Theorem 1. For any number but and any natural n. and k. Equality is true:
Different: if but - any number; n. and k. Natural numbers, then:
Hence Rule 1:
3. Explanatory tasks
Output: Private cases confirmed the correctness of Theorem No. 1. We prove it in the general case, that is, for any but and any natural n. and k.
4. Proof of Theorem 1
Dano number but - anyone; numbers n. and k - Natural. Prove
The proof is based on determining the degree.
5. Solution of examples with the theorem 1
Example 1: Imagine in the form of degree.
To solve the following examples, we use the theorem 1.
g) 
6. Generalization of Theorem 1
The generalization is used here:
7. Solution of examples with the help of generalization of Theorem 1
8. Solution of various tasks using Theorem 1
Example 2: Calculate (you can use the basket of main degrees).
but) 
(on the table)
b) 
Example 3: Write down in the form of a degree with a base 2.
but) 
Example 4: Determine the sign of the number:
, but - Negative, since the indicator at -13 is odd.
Example 5: Replace (·) the degree of number with the base r:
We have, that is.
9. Summing up
1. Dorofeyev G.V., Suvorova S.B., Baynovich E.A. And others. Algebra 7. 6 Edition. M.: Enlightenment. 2010
1. School assistant (source).
1. Imagine in the form of a degree:
a B C D E) 
3. Write down in the form of a degree with a base 2:
4. Determine the number of:
but) 
5. Replace (·) the degree of number with the base r:
a) R 4 · (·) \u003d R 15; b) (·) · R 5 \u003d R 6
Multiplication and division of degrees with the same indicators
In this lesson, we will study the multiplication of degrees with the same indicators. First, let us recall the basic definitions and theorems on multiplication and dividing degrees with the same bases and erend the degree to the degree. Then we formulate and prove the theorem on multiplication and dividing degrees with the same indicators. And then with their help, we decide a number of typical tasks.
Reminder basic definitions and theorems
Here a. - the foundation of the degree
— n.The degree of date.
Theorem 1. For any number but and any natural n. and k. Equality is true:
When multiplying degrees with the same bases, the indicators are folded, the base remains unchanged.
Theorem 2. For any number but and any natural n. and k, such that n. > k. Equality is true:
When dividing degrees with the same bases, the indicators are torn, and the base remains unchanged.
Theorem 3. For any number but and any natural n. and k. Equality is true:
All listed theorems were about degrees with the same basins, in this lesson will be considered degrees with the same indicators.
Examples for multiplication of degrees with the same indicators
Consider the following examples:
Cut expressions to determine the degree.
Output: From examples you can see that 
but it still needs to prove. We formulate the theorem and prove it in the general case, that is, for any but and b. and any natural n.
The wording and proof of Theorem 4
For any numbers but and b. and any natural n. Equality is true:
Evidence Theorems 4. .
By definition of degree:
So, we proved that .
In order to multiply the degrees with the same indicators, it is sufficient to multiply the bases, and the degree indicator is unchanged.
The wording and proof of Theorem 5
We formulate the theorem for dividing degrees with the same indicators.
For any number but and b () and any natural n. Equality is true:
Evidence Theorems 5. .
Sick and by definition of the degree:
The wording by theorem of the words
So, we have proven that.
To divide each other with the same indicators, it is sufficient to divide one base to another, and the degree indicator is unchanged.
Solution of typical tasks using Theorem 4
Example 1: Present in the form of a piece of degrees.
To solve the following examples, we use theorem 4.
To solve the following example, let us recall the formula:
Generalization of Theorem 4.
Generalization of Theorem 4:
Solution of examples with the help of generalized theorem 4
Continued solving typical tasks
Example 2: Write down in the form of a degree of work.
Example 3: Write down in the form of degree with an indicator 2.
Examples for calculation
Example 4: Calculate the most rational way.
2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: Ventana Graph
3. Kolyagin Yu.M., Tkachev M.V., Fedorova N.E. and others. Algebra 7 .m .: Enlightenment. 2006
2. School assistant (source).
1. Present in the form of a work of degrees:
but) ; b); at) ; d);
2. Record the work as a degree:
3. Write down in the form of a degree with an indicator 2:
4. Calculate the most rational way.
Mathematics lesson on the topic "Multiplication and division of degrees"
Sections: Mathematics
Pedagogical goal:
Tasks:
Activity units of exercises: determination of a natural indicator; degree components; definition of private; The combination law of multiplication.
I. Organization of demonstration mastering students' knowledge. (step 1)
a) actualization of knowledge:
2) formulate the determination of the degree with the natural indicator.
a n \u003d a a a a A ... A (n times)
b k \u003d b b b b a ... b (k times) justify the answer.
II. Organization of self-examination of the trained degree of ownership of current experience. (step 2)
Test for self-test: (individual work in two versions.)
A1) Prepare a piece of 7 7 7 7 x x x in the form of a degree:
A2) Present in the form of a product (-3) 3 x 2
A3) Calculate: -2 3 2 + 4 5 3
The number of tasks in the test I select in accordance with the preparation of the class level.
To the test I give a key for self-test. Criteria: Start - not standing.
III. Educational and practical task (step 3) + Step 4. (Formulate the properties of the students themselves)
During the solution of the problem 1) and 2), students offer a decision, and I, as a teacher, organizing a class on finding a method for simplifying degrees when multiplying with the same bases.
Teacher: Invent a way to simplify degrees when multiplying with the same bases.
An entry appears on the cluster:
The topic of the lesson is formulated. Multiplication of degrees.
Teacher: Invent the degree division rule with the same bases.
Reasoning: What action is the division is checked? A 5: A 3 \u003d? that a 2 a 3 \u003d a 5
Returning to the scheme - the cluster and complement the recording - ... deduction by subtract and add the topic of the lesson. ... and dividing degrees.
IV. Message to students of the limits of knowledge (at least and as a maximum).
Teacher: The task of a minimum for today's lesson is to learn to apply the properties of multiplication and dividing degrees with the same bases, and the maximum: apply multiplication and division together.
On the board are written : a m and n \u003d a m + n; A M: A N \u003d A M-N
V. Organization of the study of a new material. (step 5)
a) on the textbook: №403 (A, B, D) tasks with different wording
№404 (A, D, E) Independent work, then organize a mutual test, I give keys.
b) With what value m is equality? a 16 a m \u003d a 32; x H x 14 \u003d x 28; x 8 (*) \u003d x 14
Task: come up with similar examples for division.
c) № 417 (a), №418 (a) Traps for students: x 3 x n \u003d x 3n; 3 4 3 2 \u003d 9 6; A 16: A 8 \u003d A 2.
Vi. Generalization of the studied, conducting diagnostic work (which encourages students, and not teachers study this topic) (step 6)
Diagnostic work.
Test (Put keys on the back of the test).
Object options: Present in the form of a degree of private x 15: x 3; Prepare the product (-4) 2 (-4) 5 (-4) 7; With which M is equality a 16 A m \u003d A 32; Find the value of the expression H 0: H 2 at h \u003d 0.2; Calculate the expression value (5 2 5 0): 5 2.
The outcome of the lesson. Reflection. I share a class into two groups.
Find the arguments of the I Group: in favor of the knowledge of the properties of the degree, and the II group are arguments that will say that you can do without properties. All the answers are listening, we draw conclusions. In subsequent lessons, it is possible to offer statistical data and call the rubric "does not fit in the head!"
VII. Homework.
Historical reference. What numbers are called farm numbers.
P.19. №403, №408, №417
Used Books:
Properties of degrees, wording, evidence, examples.
After the number of the number is determined, it is logical to talk about properties of degree. In this article we will give the basic properties of the degree of the number, while tamping all possible degree rates. Here we also give the evidence of all the properties of the degree, as well as show how these properties apply when solving examples.
Navigating page.
Properties of degrees with natural indicators
By determining the degree with a natural indicator, the degree of a n is a product of N multipliers, each of which is a. Pushing out this definition, as well as using properties multiplying valid numbers, you can get and justify the following properties of the degree with a natural indicator:
- if a\u003e 0, then a n\u003e 0 for any natural N;
- if a \u003d 0, then a n \u003d 0;
- if a 2 · m\u003e 0, if a 2 · m-1 n;
- if m and n are such natural numbers that M\u003e n, then at 0m n, and at a\u003e 0 is true inequality a m\u003e a n.
- a m · a n \u003d a m + n;
- a m: a n \u003d a m-n;
- (a · b) n \u003d a n · b n;
- (A: b) n \u003d a n: b n;
- (a m) n \u003d a m · n;
- if n is an integer positive number, a and b - positive numbers, and A n n and a -n\u003e b -n;
- if M and N are integers, and M\u003e n, then at 0m n, and at a\u003e 1, the inequality A M\u003e A N is performed.
Immediately note that all recorded equality are identical When complying with these conditions, and their right and left parts can be changed in places. For example, the main property of fractions a m · a n \u003d a m + n simplify expressions It is often used as a m + n \u003d a m · a n.
Now consider each of them in detail.
Let's start with the properties of the work of two degrees with the same bases called the main property of degree: For any actual number A and any natural numbers M and N, the equality A m · a n \u003d a m + n is valid.
We prove the basic property of the degree. By definition of degree with a natural indicator, the product of degrees with the same bases of the form A M · A n can be written as a work 
. By virtue of the multiplication properties, the expression obtained can be written as 
, And this product is the degree of number A with a natural indicator M + n, that is, a m + n. This is the proof completed.
Let us give an example confirming the basic property of the degree. Take degrees with the same bases 2 and natural degrees 2 and 3, according to the main property of the degree, you can record the equality 2 2 · 2 3 \u003d 2 2 + 3 \u003d 2 5. Check its justice, for which I calculate the values \u200b\u200bof expressions 2 2 · 2 3 and 2 5. Performing the construction of the extent, we have 2 2 · 2 3 \u003d (2 · 2) · (2 \u200b\u200b· 2 · 2) \u003d 4 · 8 \u003d 32 and 2 5 \u003d 2 · 2 · 2 · 2 · 2 \u003d 32, as it turns out Equal values, then equality 2 2 · 2 3 \u003d 2 5 - correct, and it confirms the main property of the degree.
The main property of the degree based on the properties of multiplication can be generalized on the work of three and more degrees with the same bases and natural indicators. So for any number K natural numbers N 1, N 2, ..., n k, the equality A n 1 · a n 2 · ... · a n k \u003d a n 1 + n 2 + ... + n k.
For example, (2.1) 3 · (2.1) 3 · (2,1) 4 · (2,1) 7 \u003d (2.1) 3 + 3 + 4 + 7 \u003d (2,1) 17.
You can move to the following property of degrees with a natural indicator - property of private degrees with the same grounds: For any different number of valid number A and arbitrary natural numbers M and N, satisfying the M\u003e N condition, the equality A M: A n \u003d A M-N is true.
Before bringing the proof of this property, we will discuss the meaning of additional conditions in the wording. Condition A ≠ 0 is necessary in order to avoid dividing to zero, as 0 n \u003d 0, and when you meet the division, we agreed that it is impossible to divide to zero. The M\u003e n condition is introduced so that we do not go beyond the scope of the natural indicators. Indeed, at m\u003e n, the degree of Am-N is a natural number, otherwise it will be either zero (which is happening at M-N) or a negative number (which is happening at MM-N · AN \u003d A (M-N) + N \u003d am. From the obtained equality Am-n · An \u003d AM and from the connection of multiplication with the division, it follows that AM-N is private degrees AM and AN. This is proved by the property of private degrees with the same bases.
Let us give an example. Take two degrees with the same bases π and natural indicators 5 and 2, the considered degree of the degree corresponds to the equality π 5: π 2 \u003d π 5-3 \u003d π 3.
Now consider property of a work: The natural degree n of the work of two any real numbers a and b is equal to the product of the degrees a n and b n, that is, (a · b) n \u003d a n · b n.
Indeed, by determining the degree with a natural indicator we have 
. The last work on the basis of multiplication properties can be rewritten as 
As equal to a n · b n.
Let us give an example: 
.
This property extends to the degree of product of three and more multipliers. That is, the property of natural degree n The works of the multipliers are recorded as (a 1 · a 2 · ... · a k) n \u003d a 1 n · a 2 n · ... · a k n.
For clarity, we will show this property on the example. For the work of three factors to the degree 7 we have.
The following property is private property in kind: Private valid numbers a and b, b ≠ 0 to natural degree n is equal to private degrees a n and b n, that is, (a: b) n \u003d a n: b n.
Proof can be carried out using the previous property. So (a: b) n · bn \u003d ((a: b) · b) n \u003d an, and from equality (a: b) n · bn \u003d An follows that (a: b) n is private from dividing an on BN.
We write this property on the example of specific numbers: 
.
Now voiced degree in degree: For any actual number A and any natural numbers M and N, the degree of a m to the degree n is equal to the degree of number A with an indicator M · n, that is, (a m) n \u003d a m · n.
For example, (5 2) 3 \u003d 5 2 · 3 \u003d 5 6.
Proof of the degree property of the degree is the following chain of equalities: 
.
The considered property can be extended to a degree to degree to degree, etc. For example, for any natural numbers P, Q, R and S. Equality is fair 
. For greater clarity, we give an example with specific numbers: (((5.2) 3) 2) 5 \u003d (5.2) 3 + 2 + 5 \u003d (5.2) 10.
It remains to dwell on the properties of the comparison of degrees with a natural indicator.
Let's start with the proof of the properties of the zero comparison and the degree with the natural indicator.
For a start, we justify that a n\u003e 0 for any A\u003e 0.
The product of two positive numbers is a positive number that follows from the definition of multiplication. This fact and multiplication properties suggest that the result of multiplying any number of positive numbers will also be a positive number. And the degree of number A with a natural indicator N by definition is a product of N multipliers, each of which is a. These arguments suggest that for any positive base a degree a n there is a positive number. By virtue of the proven property 3 5\u003e 0, (0.00201) 2\u003e 0 and 
.
It is quite obvious that for any natural N at a \u003d 0 degree a n is zero. Indeed, 0 n \u003d 0 · 0 · ... · 0 \u003d 0. For example, 0 3 \u003d 0 and 0 762 \u003d 0.
Go to the negative foundations of the degree.
Let's start with the case when the degree indicator is an even number, we denote it as 2 · m, where M is natural. Then 
. According to the rule of multiplication of negative numbers, each of the works of the form A · A is equal to the product of the modules of the numbers a and a, which means it is a positive number. Consequently, the work will be positive 
and degree a 2 · m. We give examples: (-6) 4\u003e 0, (-2.2) 12\u003e 0 and.
Finally, when the base of degree A is a negative number, and the indicator of the degree is an odd number 2 · m-1, then 
. All works A · A are positive numbers, the product of these positive numbers is also positive, and its multiplication to the remaining negative number A as a result of a negative number. By virtue of this property (-5) 3 17 N n, it is a product of the left and right parts of n faithful inequalities A the properties of inequalities are fair and proved inequality of the form a n n. For example, due to this property, inequalities 3 7 7 are valid 
.
It remains to prove the last of the listed properties of degrees with natural indicators. Word it. Of two degrees with natural indicators and the same positive grounds that are smaller than the units, the larger one is less than; And of two degrees with natural indicators and the same bases, large units, more than the degree, the indicator of which is greater. Go to the proof of this property.
We prove that at m\u003e n and 0m n. To do this, we write the difference A M-N and compare it with zero. Recorded difference after making a n per brackets will take the form a n · (a m-n -1). The resulting product is negative as a product of the positive number of AN and the negative number AM-N -1 (AN is positive as the natural degree of a positive number, and the difference Am-n -1 is negative, since M-n\u003e 0 is due to the original condition M\u003e N, where It follows that at 0m-n less than one). Consequently, a M -a n m n, which was required to prove. For example, we give the faithful inequality.
It remains to prove the second part of the property. We prove that at m\u003e n and a\u003e 1, a m\u003e a n is true. The difference A M -A n after making a n for parentheses takes the form a n · (a m-n -1). This product is positive, since at a\u003e 1 degree an AN there is a positive number, and the difference AM-N -1 is a positive number, since M-n\u003e 0 due to the initial condition, and at a\u003e 1 degree of am-n more units . Consequently, a M -a n\u003e 0 and a m\u003e a n, which was required to prove. The illustration of this property serves inequality 3 7\u003e 3 2.
Properties of degrees with integer indicators
Since the entire positive numbers are natural numbers, then all the properties of degrees with integer positive indicators exactly coincide with the properties of degrees with natural indicators listed and proven in the previous paragraph.
The degree with a whole negative indicator, as well as the degree with the zero indicator, we determined so that all the properties of the degrees with natural indicators are valid, expressed by equalities. Therefore, all these properties are valid for zero degree, and for negative indicators, while, of course, the bases of degrees are different from zero.
So, for any valid and different numbers of numbers a and b, as well as any integers M and N are the following properties of degrees with integer indicators:
At a \u003d 0 degrees a m and a n, it makes sense only when M, and N positive integers, that is, natural numbers. Thus, the newly recorded properties are also valid for cases when a \u003d 0, and the numbers M and N are integers positive.
It is not difficult to prove each of these properties, it is enough to use the definitions of the degree with a natural and integer, as well as the properties of actions with valid numbers. For example, we prove that the degree property is performed both for entire positive numbers and for integral numbers. To do this, it is necessary to show that if P is zero or a natural number and q is zero or a natural number, then equality (AP) Q \u003d AP · Q, (A -P) Q \u003d A (-p) · Q, (AP ) -Q \u003d ap · (-q) and (a -p) -q \u003d a (-p) · (-q). Let's do it.
For positive p and q, equality (A p) Q \u003d A p · Q is proved in the previous paragraph. If p \u003d 0, then we have (a 0) q \u003d 1 q \u003d 1 and a 0 · q \u003d a 0 \u003d 1, from where (a 0) q \u003d a 0 · q. Similarly, if Q \u003d 0, then (a p) 0 \u003d 1 and a p · 0 \u003d a 0 \u003d 1, from where (a p) 0 \u003d a p · 0. If, and p \u003d 0 and q \u003d 0, then (a 0) 0 \u003d 1 0 \u003d 1 and a 0 · 0 \u003d a 0 \u003d 1, from where (a 0) 0 \u003d a 0 · 0.
Now we prove that (a -p) q \u003d a (-p) · q. To determine the degree with a whole negative indicator, then 
. By the property of private to the extent we have 
. Since 1 p \u003d 1 · 1 · ... · 1 \u003d 1 and, then. The last expression by definition is the degree of type A - (p · q), which, by virtue of the multiplication rules, can be written as a (-p) · q.
Similarly 
.
AND 
.
By the same principle, you can prove all other properties of the degree with an integer recorded in the form of equalities.
In the penultimate of the recorded properties, it is worth staying on the proof of A -N\u003e B -N inequality, which is valid for any whole negative -N and any positive A and B, for which condition A is satisfied . We write and transform the difference between the left and right parts of this inequality: 
. As under condition a n n, therefore, b n -a n\u003e 0. The product A n · b n is also positive as a product of positive numbers a n and b n. Then the resulting fraction is positive as the private positive numbers b n-n and a n · b n. Therefore, from where a -n\u003e b -n, which was required to prove.
The last property of degrees with integer indicators is proved in the same way as a similar property of degrees with genuine indicators.
The properties of degrees with rational indicators
We determined the degree with a fractional indicator by spreading the properties of the degree with an integer. In other words, the degrees with fractional indicators have the same properties as degrees with integer indicators. Namely:
- property of the work of degrees with the same bases
at a\u003e 0, and if, then at a≥0; - private degree property with identical grounds
at a\u003e 0; - property of the work in fractional degree
at a\u003e 0 and b\u003e 0, and if, at a≥0 and (or) b≥0; - private property in fractional degree
at a\u003e 0 and b\u003e 0, and if, at a≥0 and b\u003e 0; - degree
at a\u003e 0, and if, then at a≥0; - the comparison of degrees with equal rational indicators: for any positive numbers a and b, a 0 Fairly inequality A p p, and at p p\u003e b p;
- the properties of comparison of degrees with rational indicators and equal bases: for rational numbers p and q, p\u003e q at 0p q, and at a\u003e 0 - inequality a p\u003e a q.
- a p · a q \u003d a p + q;
- a P: A Q \u003d A P-Q;
- (a · b) p \u003d a p · b p;
- (A: b) p \u003d a p: b p;
- (A p) Q \u003d A p · Q;
- for any positive numbers a and b, a 0 Fairly inequality A p p, and at p p\u003e b p;
- for irrational numbers p and q, p\u003e q at 0p q, and at a\u003e 0 - inequality a p\u003e a q.
Proof of the properties of degrees with fractional indicators is based on determining the degree with a fractional indicator, on the properties of the arithmetic root of N-essential and on the degree properties with an integer. We give proofs.
To determine the degree with a fractional indicator and, then 
. The properties of the arithmetic root allow us to write down the following equalities. Next, using the property of the degree with the integer, we get from where to determine the degree with a fractional indicator we have 
And the indicator of the degree obtained can be transformed as follows :. This is the proof completed.
Absolutely similarly proves the second property of degrees with fractional indicators: 
For similar principles, the rest of the equality are proved: 
Go to the proof of the next property. We prove that for any positive a and b, a 0 Fairly inequality A p p, and at p p\u003e b p. We write the rational number P as M / N, where M is an integer, and N is natural. Conditions P 0 in this case will be equivalent to the conditions M 0, respectively. With m\u003e 0 and am m. From this inequality to the properties of the roots we have, and since A and B are positive numbers, then on the basis of determining the degree with a fractional indicator, the obtained inequality can be rewritten as, that is, a p p.
Similarly, at m m\u003e b m, from where, that is, and a p\u003e b p.
It remains to prove the last of the listed properties. We prove that for rational numbers p and q, p\u003e q at 0p q, and at a\u003e 0 - inequality a p\u003e a q. We can always lead to a common denominator rational numbers p and q, even if we get ordinary fractions and, where m 1 and m 2 are integers, and N is natural. In this case, the condition p\u003e Q will correspond to the condition M 1\u003e M 2, which follows from the rule of comparison of ordinary fractions with the same denominators. Then, according to the property of comparison of degrees with the same bases and natural indicators at 0m 1 m 2, and at a\u003e 1 - inequality A M 1\u003e A M 2. These inequalities on the properties of the roots can be rewritten according to 
and 
. And the determination of the degree with a rational indicator allows you to move to inequalities and, accordingly. From here we make the final conclusion: at p\u003e q and 0p q, and at a\u003e 0 - inequality a p\u003e a q.
Properties of degrees with irrational indicators
From how the degree with an irrational indicator is determined, it can be concluded that it has all the properties of degrees with rational indicators. So for any a\u003e 0, b\u003e 0 and irrational numbers p and q are the following properties of degrees with irrational indicators:
From here, we can conclude that the degrees with any valid parameters p and q at a\u003e 0 have these same properties.
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First level
The degree and properties. Exhaustive Guide (2019)
Why are you needed? Where will they come to you? Why do you need to spend time on their study?
To find out all about degrees, what they need about how to use their knowledge in everyday life read this article.
And, of course, the knowledge of degrees will bring you closer to the successful surrender of OGE or the EGE and to enter the university of your dreams.
Let "S GO ... (drove!)
Important remark! If instead of formulas you see abracadabra, clean the cache. To do this, click Ctrl + F5 (on Windows) or CMD + R (on Mac).
FIRST LEVEL
The exercise is the same mathematical operation as addition, subtraction, multiplication or division.
Now I will explain all the human language on very simple examples. Be careful. Examples of elementary, but explaining important things.
Let's start with addition.
There is nothing to explain here. You all know everything: we are eight people. Everyone has two bottles of cola. How much is the cola? Right - 16 bottles.
Now multiplication.
The same example with a cola can be recorded differently :. Mathematics - People cunning and lazy. They first notice some patterns, and then invent the way how to "count" them faster. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a reception called multiplication. Agree, it is considered easier and faster than.
So, to read faster, easier and without mistakes, you just need to remember table multiplication. Of course, you can do everything more slowly, harder and mistakes! But…
Here is the multiplication table. Repeat.
And the other, more beautiful:
And what other tricks came up with lazy mathematicians? Right - erection.
Erection
If you need to multiply the number for yourself five times, then mathematics say that you need to build this number in the fifth degree. For example, . Mathematics remember that two in the fifth degree is. And they solve such tasks in the mind - faster, easier and without errors.
For this you need only remember what is highlighted in color in the table of degrees of numbers. Believe it, it will greatly facilitate your life.
By the way, why the second degree is called square numbers, and the third - cuba? What does it mean? Very good question. Now there will be to you and squares, and Cuba.
Example from life number 1
Let's start with a square or from a second degree of number.
Imagine a square pool of meter size on a meter. The pool is on your dacha. Heat and really want to swim. But ... Pool without the bottom! You need to store the bottom of the pool tiles. How much do you need tiles? In order to determine this, you need to find out the area of \u200b\u200bthe bottom of the pool.
You can simply calculate, with a finger, that the bottom of the pool consists of a meter cubes per meter. If you have a meter tile for meter, you will need to pieces. It's easy ... But where did you see such a tile? The tile is more likely to see for see and then "finger to count" torture. Then you have to multiply. So, on one side of the bottom of the pool, we fit tiles (pieces) and on the other too tiles. Multiplying on, you will get tiles ().
Did you notice that in order to determine the area of \u200b\u200bthe bottom of the pool, did we multiply the same number by yourself? What does it mean? This is multiplied by the same number, we can take advantage of the "erection of the extermination". (Of course, when you have only two numbers, multiply them or raise them into the degree. But if you have a lot of them, it is much easier to raise them in terms of calculations, too much less. For the exam, it is very important).
So thirty to the second degree will (). Or we can say that thirty in the square will be. In other words, the second degree of number can always be represented as a square. And on the contrary, if you see a square - it is always the second degree of some number. Square is the image of a second degree number.
Example from life number 2
Here is the task, count how many squares on a chessboard with a square of the number ... on one side of the cells and on the other too. To calculate their quantity, you need to multiply eight or ... If you note that the chessboard is a square of the side, then you can build eight per square. It turns out cells. () So?
Example from life number 3
Now a cube or a third degree of number. The same pool. But now you need to know how much water will have to fill in this pool. You need to count the volume. (Volumes and liquids, by the way, are measured in cubic meters. Suddenly, really?) Draw a pool: bottom of the meter size and a depth of meters and try to count how much cubes the size of the meter on the meter will enter your pool.
Right show your finger and count! Once, two, three, four ... twenty two, twenty three ... how much did it happen? Did not come down? Difficult to count your finger? So that! Take an example from mathematicians. They are lazy, therefore noticed that to calculate the volume of the pool, it is necessary to multiply each other in length, width and height. In our case, the volume of the pool will be equal to cubes ... it is easier for the truth?
And now imagine, as far as Mathematics are lazy and cunning, if they are simplified. Brought all to one action. They noticed that the length, width and height is equal to and that the same number varnims itself on itself ... And what does this mean? This means that you can take advantage of the degree. So, what did you think with your finger, they do in one action: three in Cuba is equal. This is written so :.
It remains only remember Table degrees. If you are, of course, the same lazy and cunning as mathematics. If you like to work a lot and make mistakes - you can continue to count your finger.
Well, to finally convince you that the degrees came up with Lodii and cunnies to solve their life problems, and not to create problems you, here's another couple of examples from life.
Example from life number 4
You have a million rubles. At the beginning of each year you earn every million another million. That is, every million will double at the beginning of each year. How much money will you have in the years? If you are sitting now and "you think your finger", then you are a very hardworking person and .. stupid. But most likely you will answer in a couple of seconds, because you are smart! So, in the first year - two multiplied two ... in the second year - what happened, another two, on the third year ... Stop! You noticed that the number multiplies itself. So, two in the fifth degree - a million! And now imagine that you have a competition and these million will receive the one who will find faster ... It is worth remembering the degree of numbers, what do you think?
Example from life number 5
You have a million. At the beginning of each year you earn each million two more. Great truth? Every million triples. How much money will you have after a year? Let's count. The first year is to multiply on, then the result is still on ... already boring, because you have already understood everything: three is multiplied by itself. Therefore, the fourth degree is equal to a million. It is only necessary to remember that three in the fourth degree is or.
Now you know that with the help of the erection of the number, you will greatly facilitate your life. Let's look next to what you can do with the degrees and what you need to know about them.
Terms and concepts ... so as not to get confused
So, for starters, let's define the concepts. What do you think, what is the indicator of the degree? It is very simple - this is the number that is "at the top" of the degree of number. Not scientifically, but it is clear and easy to remember ...
Well, at the same time that such a foundation degree? Even easier - this is the number that is below, at the base.
Here is a drawing for loyalty.
Well, in general, to summarize and better remember ... The degree with the basis "" and the indicator "" is read as "to degree" and is written as follows:
The degree of number with a natural indicator
You already probably guessed: because the indicator is a natural number. Yes, but what is natural number? Elementary! Natural These are the numbers that are used in the account when listing items: one, two, three ... We, when we consider items, do not say: "Minus five", "minus six", "minus seven". We also do not say: "one third", or "zero of whole, five tenths." These are not natural numbers. And what these numbers do you think?
Numbers like "minus five", "minus six", "minus seven" belong to whole numbers. In general, to whole numbers include all natural numbers, the numbers are opposite to natural (that is, taken with a minus sign), and the number. Zero understand easily - this is when nothing. And what do they mean negative ("minus") numbers? But they were invented primarily to designate debts: if you have a balance on the phone number, it means that you should operator rubles.
All sorts of fractions are rational numbers. How did they arise, what do you think? Very simple. Several thousand years ago, our ancestors found that they lack natural numbers to measure long, weight, square, etc. And they invented rational numbers... I wonder if it's true?
There are also irrational numbers. What is this number? If short, then an infinite decimal fraction. For example, if the circumference length is divided into its diameter, then the irrational number will be.
Summary:
We define the concept of degree, the indicator of which is a natural number (i.e., a whole and positive).
- Any number to the first degree equally to itself:
- Evaluate the number in the square - it means to multiply it by itself:
- Evaluate the number in the cube - it means to multiply it by itself three times:
Definition. Evaluate the number in a natural degree - it means to multiply the number of all time for yourself:
.
Properties of degrees
Where did these properties come from? I will show you now.
Let's see: what is and ?
By definition:
How many multipliers are here?
Very simple: we completed multipliers to multipliers, it turned out the factors.
But by definition, this is the degree of a number with an indicator, that is, that, that it was necessary to prove.
Example: Simplify the expression.
Decision:
Example: Simplify the expression.
Decision: It is important to notice that in our rule before Must be the same foundation!
Therefore, we combine degrees with the basis, but remains a separate multiplier:
only for the work of degrees!
In no case can not write that.
2. That is The degree of number
Just as with the previous property, we turn to the definition of the degree:
It turns out that the expression is multiplied by itself once, that is, according to the definition, this is, there is a number of number:
In fact, this can be called "the indicator for brackets". But never can do it in the amount:
Recall the formula of abbreviated multiplication: how many times did we want to write?
But it is incorrect, because.
Negative
Up to this point, we only discussed what the indicator should be.
But what should be the basis?
In the degrees of S. natural indicator The base can be any number. And the truth, we can multiply each other any numbers, whether they are positive, negative, or even.
Let's think about what signs ("or" ") will have the degrees of positive and negative numbers?
For example, a positive or negative number? BUT? ? With the first, everything is clear: how many positive numbers we are not multiplied by each other, the result will be positive.
But with negative a little more interesting. After all, we remember a simple rule of grade 6: "Minus for minus gives a plus." That is, or. But if we multiply on, it will work out.
Determine independently, what sign the following expressions will have:
| 1) | 2) | 3) |
| 4) | 5) | 6) |
Cope?
Here are the answers: in the first four examples, I hope everything is understandable? Just look at the base and indicator, and apply the appropriate rule.
1) ; 2) ; 3) ; 4) ; 5) ; 6) .
In example 5), everything is also not as scary, as it seems: it doesn't matter what is equal to the base - the degree is even, which means that the result will always be positive.
Well, with the exception of the case when the base is zero. The reason is not equal? Obviously no, because (because).
Example 6) is no longer so simple!
6 Examples for Training
Solutions of 6 examples
If you do not pay attention to the eighth degree, what do we see here? Remember the Grade 7 program. So, remembered? This is a formula for abbreviated multiplication, namely - the difference of squares! We get:
Carefully look at the denominator. He is very similar to one of the multipliers of the numerator, but what's wrong? Not the procedure of the terms. If they would change them in places, it would be possible to apply the rule.
But how to do that? It turns out very easy: the even degree of denominator helps us.
Magically, the components changed in places. This "phenomenon" is applicable for any expression to an even degree: we can freely change signs in brackets.
But it is important to remember: all signs are changing at the same time.!
Let's go back for example:
And again the formula:
Integer We call natural numbers opposite to them (that is, taken with the sign "") and the number.
whole positive number, And it does not differ from natural, then everything looks exactly as in the previous section.
And now let's consider new cases. Let's start with an indicator equal to.
Any number to zero equal to one:
As always, we will ask me: why is it so?
Consider any degree with the basis. Take, for example, and domineering on:
So, we multiplied the number on, and got the same as it was. And for what number must be multiplied so that nothing has changed? That's right on. So.
We can do the same with an arbitrary number:
Repeat the rule:
Any number to zero equal to one.
But from many rules there are exceptions. And here it is also there is a number (as a base).
On the one hand, it should be equal to any extent - how much zero itself is neither multiplied, still get zero, it is clear. But on the other hand, like any number to zero degree, should be equal. So what's the truth? Mathematics decided not to bind and refused to erect zero to zero. That is, now we can not only be divided into zero, but also to build it to zero.
Let's go further. In addition to natural numbers and numbers include negative numbers. To understand what a negative degree, we will do as last time: Domingly some normal number on the same to a negative degree:
From here it is already easy to express the desired:
Now we spread the resulting rule to an arbitrary degree:
So, we formulate the rule:
The number is a negative degree back to the same number to a positive degree. But at the same time the base can not be zero: (Because it is impossible to divide).
Let's summarize:
I. The expression is not defined in the case. If, then.
II. Any number to zero is equal to one :.
III. A number that is not equal to zero, to a negative degree back to the same number to a positive degree :.
Tasks for self solutions:
Well, as usual, examples for self solutions:
Task analysis for self solutions:
I know, I know, the numbers are terrible, but the exam should be ready for everything! Share these examples or scatter their decision, if I could not decide and you will learn to easily cope with them on the exam!
Continue expanding the circle of numbers, "suitable" as an indicator of the degree.
Now consider rational numbers. What numbers are called rational?
Answer: All that can be represented in the form of fractions, where and - integers, and.
To understand what is "Freight degree", Consider the fraction:
Erected both parts of the equation to the degree:
Now remember the rule about "Degree to degree":
What number should be taken to the degree to get?
This formulation is the definition of root degree.
Let me remind you: the root of the number () is called the number that is equal in the extermination.
That is, the root degree is an operation, reverse the exercise into the degree :.
It turns out that. Obviously, this particular case can be expanded :.
Now add a numerator: what is? The answer is easy to get with the help of the "degree to degree" rule:
But can the reason be any number? After all, the root can not be extracted from all numbers.
No one!
Remember the rule: any number erected into an even degree is the number positive. That is, to extract the roots of an even degree from negative numbers it is impossible!
This means that it is impossible to build such numbers into a fractional degree with an even denominator, that is, the expression does not make sense.
What about expression?
But there is a problem.
The number can be represented in the form of DRGIH, reduced fractions, for example, or.
And it turns out that there is, but does not exist, but it's just two different records of the same number.
Or another example: once, then you can write. But it is worthwhile to write to us in a different way, and again we get a nuisance: (that is, they received a completely different result!).
To avoid similar paradoxes, we consider only a positive foundation of degree with fractional indicator.
So, if:
- - natural number;
- - integer;
Examples:
The degrees with the rational indicator are very useful for converting expressions with roots, for example:
5 examples for training
Analysis of 5 examples for training
Well, now - the most difficult. Now we will understand irrational.
All the rules and properties of degrees here are exactly the same as for a degree with a rational indicator, with the exception
After all, by definition, irrational numbers are numbers that cannot be represented in the form of a fraction, where and - integers (that is, irrational numbers are all valid numbers except rational).
When studying degrees with natural, whole and rational indicator, we each time constituted a certain "image", "analogy", or a description in more familiar terms.
For example, a natural figure is a number, several times multiplied by itself;
...zero - this is how the number multiplied by itself once, that is, it has not yet begun to multiply, it means that the number itself has not even appeared - therefore the result is only a certain "billet number", namely the number;
...degree with a whole negative indicator "It seemed to have occurred a certain" reverse process ", that is, the number was not multiplied by itself, but Deli.
By the way, in science is often used with a complex indicator, that is, the indicator is not even a valid number.
But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the Institute.
Where we are sure you will do! (If you learn to solve such examples :))
For example:
Solim yourself:
Debris:
1. Let's start with the usual rules for the exercise rules for us:
Now look at the indicator. Doesn't he remind you of anything? Remember the formula of abbreviated multiplication. Square differences:
In this case,
It turns out that:
Answer: .
2. We bring the fraction in the indicators of degrees to the same form: either both decimal or both ordinary. We obtain, for example:
Answer: 16.
3. Nothing special, we use the usual properties of degrees:
ADVANCED LEVEL
Determination of degree
The degree is called the expression of the form: where:
- — degree basis;
- - Indicator.
The degree with the natural indicator (n \u003d 1, 2, 3, ...)
Build a natural degree n - it means multiplying the number for yourself once:
The degree with the integer (0, ± 1, ± 2, ...)
If an indicator of the degree is software positive number:
Construction in zero degree:
The expression is indefinite, because, on the one hand, to any extent, it is, and on the other - any number of in degree is.
If an indicator of the degree is a whole negative number:
(Because it is impossible to divide).
Once again about zeros: the expression is not defined in the case. If, then.
Examples:
Rational
- - natural number;
- - integer;
Examples:
Properties of degrees
To make it easier to solve problems, let's try to understand: where did these properties come from? We prove them.
Let's see: What is what?
By definition:
So, in the right part of this expression, such a work is obtained:
But by definition, this is the degree of a number with an indicator, that is:
Q.E.D.
Example : Simplify the expression.
Decision : .
Example : Simplify the expression.
Decision : It is important to notice that in our rule beforethere must be the same bases. Therefore, we combine degrees with the basis, but remains a separate multiplier:
Another important note: this is a rule - only for the work of degrees!
In no case to the nerve to write that.
Just as with the previous property, we turn to the definition of the degree:
We regroup this work like this:
It turns out that the expression is multiplied by itself once, that is, according to the definition, this is - by the degree of number:
In fact, this can be called "the indicator for brackets". But never can do this in the amount:!
Recall the formula of abbreviated multiplication: how many times did we want to write? But it is incorrect, because.
Degree with a negative basis.
Up to this point, we only discussed what should be index degree. But what should be the basis? In the degrees of S. natural indicator The base can be any number .
And the truth, we can multiply each other any numbers, whether they are positive, negative, or even. Let's think about what signs ("or" ") will have the degrees of positive and negative numbers?
For example, a positive or negative number? BUT? ?
With the first, everything is clear: how many positive numbers we are not multiplied by each other, the result will be positive.
But with negative a little more interesting. After all, we remember a simple rule of grade 6: "Minus for minus gives a plus." That is, or. But if we will multiply on (), it turns out.
And so to infinity: each time the next multiplication will change the sign. Simple rules can be formulated:
- even degree - number positive.
- Negative number erected into odd degree - number negative.
- A positive number to either degree is the number positive.
- Zero to any degree is zero.
Determine independently, what sign the following expressions will have:
| 1. | 2. | 3. |
| 4. | 5. | 6. |
Cope? Here are the answers:
1) ; 2) ; 3) ; 4) ; 5) ; 6) .
In the first four examples, I hope everything is clear? Just look at the base and indicator, and apply the appropriate rule.
In example 5), everything is also not as scary, as it seems: it doesn't matter what is equal to the base - the degree is even, which means that the result will always be positive. Well, with the exception of the case when the base is zero. The reason is not equal? Obviously no, because (because).
Example 6) is no longer so simple. Here you need to know that less: or? If you remember that it becomes clear that, and therefore, the base is less than zero. That is, we apply the rule 2: the result will be negative.
And again we use the degree of degree:
All as usual - write down the definition of degrees and, divide them to each other, divide on the pairs and get:
Before you disassemble the last rule, we solve several examples.
Calculated expressions:
Solutions :
If you do not pay attention to the eighth degree, what do we see here? Remember the Grade 7 program. So, remembered? This is a formula for abbreviated multiplication, namely - the difference of squares!
We get:
Carefully look at the denominator. He is very similar to one of the multipliers of the numerator, but what's wrong? Not the procedure of the terms. If they were swapped in places, it would be possible to apply the rule 3. But how to do it? It turns out very easy: the even degree of denominator helps us.
If you draw it on, nothing will change, right? But now it turns out the following:
Magically, the components changed in places. This "phenomenon" is applicable for any expression to an even degree: we can freely change signs in brackets. But it is important to remember: all signs are changing at the same time!You can not replace on, changing only one disagreeable minus!
Let's go back for example:
And again the formula:
So now the last rule:
How will we prove? Of course, as usual: I will reveal the concept of degree and simplifies:
Well, now I will reveal brackets. How much will the letters get? Once on multipliers - what does it remind? It is nothing but the definition of the operation multiplication: In total there were factors. That is, it is, by definition, the degree of number with the indicator:
Example:
Irrational
In addition to information about degrees for the average level, we will analyze the degree with the irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational indicator, with the exception - after all, by definition, irrational numbers are numbers that cannot be submitted in the form of a fraction, where - the integers (i.e., irrational numbers are All valid numbers besides rational).
When studying degrees with natural, whole and rational indicator, we each time constituted a certain "image", "analogy", or a description in more familiar terms. For example, a natural figure is a number, several times multiplied by itself; The number in zero degree is somehow the number multiplied by itself once, that is, it has not yet begun to multiply, it means that the number itself has not even appeared - therefore, only a certain "billet", namely, is the result; The degree with a whole negative indicator is as if a certain "reverse process" occurred, that is, the number was not multiplied by itself, but divided.
Imagine the degree with an irrational indicator is extremely difficult (just as it is difficult to submit a 4-dimensional space). It is rather a purely mathematical object that mathematics created to expand the concept of degree to the entire space of numbers.
By the way, in science is often used with a complex indicator, that is, the indicator is not even a valid number. But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the Institute.
So what do we do if we see an irrational rate? We are trying to get rid of it with all the might! :)
For example:
Solim yourself:
| 1) | 2) | 3) |
Answers:
- We remember the formula the difference of squares. Answer:.
- We give the fraction to the same form: either both decimal, or both ordinary. We get, for example:.
- Nothing special, we use the usual properties of degrees:
Summary of section and basic formulas
Degree called the expression of the form: where:
Integer
the degree, the indicator of which is a natural number (i.e., a whole and positive).
Rational
the degree, the indicator of which is negative and fractional numbers.
Irrational
the degree, the indicator of which is an infinite decimal fraction or root.
Properties of degrees
Features of degrees.
- Negative number erected into even degree - number positive.
- Negative number erected into odd degree - number negative.
- A positive number to either degree is the number positive.
- Zero to any degree is equal.
- Any number to zero equal.
Now you need a word ...
How do you need an article? Write down in the comments like or not.
Tell me about your experience in using the properties of degrees.
Perhaps you have questions. Or suggestions.
Write in the comments.
And good luck on the exams!
Lesson on the topic: "Rules for multiplication and division of degrees with the same and different indicators. Examples"
Additional materials
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Training manuals and simulators in the online store "Integral" for grade 7
Manual for textbook Yu.N. Makarychev benefit to the textbook A.G. Mordkovich
The purpose of the lesson: learns to perform actions with the degrees of the number.
To begin with, remember the concept of "degree of number". The expression of the type $ \\ underbrace (A * A * \\ ldots * a) _ (n) $ can be represented as $ a ^ n $.
It is also true inverse: $ a ^ n \u003d \\ underbrace (A * A * \\ ldots * a) _ (n) $.
This equality is called "a degree record in the form of a work." It will help us determine how to multiply and share degrees.
Remember:
a. - The foundation of the degree.
n. - Indicator.
If a n \u003d 1., So, the number but They took once and, accordingly: $ a ^ n \u003d 1 $.
If a n \u003d 0., then $ a ^ 0 \u003d 1 $.
Why this happens, we will be able to find out when we will get acquainted with the rules of multiplication and division of degrees.
Multiplication rules
a) If degrees are multiplied with the same base.To $ a ^ n * a ^ m $, write down the degree in the form of a work: $ \\ underbrace (a * a * \\ ldots * a) _ (n) * \\ underbrace (a * a * \\ ldots * a) _ (m ) $.
Figure shows that the number but have taken n + M. Once, then $ a ^ n * a ^ m \u003d a ^ (n + m) $.
Example.
$2^3 * 2^2 = 2^5 = 32$.
This property is convenient to use what to simplify work when erecting a number to a greater degree.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.
b) If degrees are multiplied with different bases, but the same indicator.
To $ a ^ n * b ^ n $, write down the degree in the form of a work: $ \\ underbrace (a * a * \\ ldots * a) _ (n) * \\ underbrace (b * b * \\ ldots * b) _ (m ) $.
If you change the multiplier places and calculate the resulting pairs, we get: $ \\ underbrace ((a * b) * (A * B) * \\ ldots * (A * B)) _ (n) $.
So, $ a ^ n * b ^ n \u003d (a * b) ^ n $.
Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.
Rules of division
a) The base of the degree is the same, different indicators.Consider dividing the degree with a high figure for dividing the degree with a smaller indicator.
So, it is necessary $ \\ FRAC (a ^ n) (a ^ m) $where n\u003e M..
We write a degree in the form of a fraction:
$ \\ FRAC (\\ underbrace (A * A * \\ ldots * a) _ (n)) (\\ underbrace (a * a * \\ ldots * a) _ (m)) $.
For convenience, the division will write in the form of a simple fraction.Now will reduce the fraction.
It turns out: $ \\ underbrace (a * a * \\ ldots * a) _ (n-m) \u003d a ^ (n-m) $.
It means $ \\ FRAC (a ^ n) (a ^ m) \u003d a ^ (n-M) $.
This property will help explain the situation with the erection of the number to the zero degree. Suppose that n \u003d M., then $ a ^ 0 \u003d a ^ (n - n) \u003d \\ FRAC (A ^ n) (a ^ n) \u003d 1 $.
Examples.
$ \\ FRAC (3 ^ 3) (3 ^ 2) \u003d 3 ^ (3-2) \u003d 3 ^ 1 \u003d 3 $.
$ \\ FRAC (2 ^ 2) (2 ^ 2) \u003d 2 ^ (2-2) \u003d 2 ^ 0 \u003d 1 $.
b) the foundation of the degree is different, the indicators are the same.
Suppose it is necessary $ \\ FRAC (A ^ n) (B ^ n) $. We write the degree of numbers in the form of a fraction:
$ \\ FRAC (\\ underbrace (A * A * \\ ldots * a) _ (n)) (\\ underbrace (b * b * \\ ldots * b) _ (n)) $.
For convenience imagine.
Using the fraction property, we break a big fraction on the work of small, we get.
$ \\ underbrace (\\ FRAC (A) (B) * \\ FRAC (A) (B) * \\ ldots * \\ FRAC (A) (B)) _ (n) $.
Accordingly, $ \\ FRAC (A ^ n) (B ^ n) \u003d (\\ FRAC (A) (B)) ^ n $.
Example.
$ \\ FRAC (4 ^ 3) (2 ^ 3) \u003d (\\ FRAC (4) (2)) ^ 3 \u003d 2 ^ 3 \u003d $ 8.
Each arithmetic operation sometimes becomes too cumbersome to record and try to simplify it. Once it was so with the operation of addition. People needed a multiple-time addition, for example, to calculate the cost of one hundred Persian carpets, the cost of which is 3 gold coins for each. 3 + 3 + 3 + ... + 3 \u003d 300. Because of the bulky, it was invented to reduce the recording to 3 * 100 \u003d 300. In fact, the recording "three multiplied to a hundred" means that you need to take a hundred trot and folded each other. Multiplication passed, gained overall popularity. But the world does not stand still, and in the Middle Ages there was a need to carry out a multiple-time multiplication. The old Indian mystery is remembered, asking for a reward for the work of wheat grain in the following quantity: for the first cell of the chessboard, he asked one grain, for the second - two, the third - four, fifth - eight and so on. Thus, the first multiplication of degrees appeared, because the amount of green was equal to the degree to the degree of the cell number. For example, on the last cell there would be 2 * 2 * 2 * ... * 2 \u003d 2 ^ 63 grain, which is equal to the number of 18 characters long, in what, in fact, the meaning of the riddles.
The exercise operation took place quite quickly, also quickly needed to carry out addition, subtraction, division and multiplication of degrees. Last and it is worth considering in more detail. Formulas for adding degrees are simple and easy to remember. In addition, it is very easy to understand where they come from if the degree is replaced by multiplication. But first should be sorted out in elementary terminology. The expression A ^ B (read "A to the degree B") means that the number A should be multiplied by itself B once, and "A" is called the foundation of the degree, and the "B" is a power indicator. If the bases of the degrees are the same, then the formulas are completely simple. Specific example: find the value of the expression 2 ^ 3 * 2 ^ 4. To know what should happen, before starting the decision to find out the answer on the computer. Having scored this expression to any online calculator, search engine, typing "multiplication of degrees with different bases the same" or a mathematical package, the output will be 128. Now we will write this expression: 2 ^ 3 \u003d 2 * 2 * 2, a 2 ^ 4 \u003d 2 * 2 * 2 * 2. It turns out that 2 ^ 3 * 2 ^ 4 \u003d 2 * 2 * 2 * 2 * 2 * 2 * 2 \u003d 2 ^ 7 \u003d 2 ^ (3 + 4). It turns out that the product of degrees with the same base is equal to the ground erected into a degree equal to the sum of the two previous degrees.
You might think that this is an accident, but no: any other example can only confirm this rule. Thus, in the general formula, the formula is as follows: a ^ n * a ^ m \u003d a ^ (n + m). There is also a rule that any number to zero equally is one. Here it is necessary to recall the rule of negative degrees: a ^ (- n) \u003d 1 / a ^ n. That is, if 2 ^ 3 \u003d 8, then 2 ^ (- 3) \u003d 1/8. Using this rule, you can prove the validity of the equality a ^ 0 \u003d 1: a ^ 0 \u003d a ^ (nn) \u003d a ^ n * a ^ (- n) \u003d a ^ (n) * 1 / a ^ (n), a ^ (n) You can reduce and the unit remains. It is also being taken out by the rule that private degrees with the same bases are equal to this base to the degree equal to the private indicator of the divide and divider: a ^ n: a ^ m \u003d a ^ (n-m). Example: Simplify the expression 2 ^ 3 * 2 ^ 5 * 2 ^ (- 7) * 2 ^ 0: 2 ^ (- 2). Multiplication is a commutative operation, therefore first the addition of multiplication indicators: 2 ^ 3 * 2 ^ 5 * 2 ^ (- 7) * 2 ^ 0 \u003d 2 ^ (3 + 5-7 + 0) \u003d 2 ^ 1 \u003d 2. Next should be dealt with the division into a negative degree. It is necessary to subtract the divider indicator from the indicator of the divide: 2 ^ 1: 2 ^ (- 2) \u003d 2 ^ (1 - (- 2)) \u003d 2 ^ (1 + 2) \u003d 2 ^ 3 \u003d 8. It turns out that the operation of division into negative The degree of identical multiplication operation to a similar positive indicator. Thus, the final answer is 8.
There are examples where there is not the canonical multiplication of degrees. Multiplying degrees with different bases is very often much more difficult, and sometimes it is impossible at all. Several examples of various possible techniques should be given. Example: Simplify the expression 3 ^ 7 * 9 ^ (- 2) * 81 ^ 3 * 243 ^ (- 2) * 729. Obviously, there is a multiplication of degrees with different bases. But it should be noted that all the foundations are different degrees of the troika. 9 \u003d 3 ^ 2.1 \u003d 3 ^ 4.3 \u003d 3 ^ 5.9 \u003d 3 ^ 6. Using the rule (a ^ n) ^ m \u003d a ^ (n * m), you should rewrite the expression in a more convenient form: 3 ^ 7 * (3 ^ 2) ^ (- 2) * (3 ^ 4) ^ 3 * ( 3 ^ 5) ^ (- 2) * 3 ^ 6 \u003d 3 ^ 7 * 3 ^ (- 4) * 3 ^ (12) * 3 ^ (- 10) * 3 ^ 6 \u003d 3 ^ (7-4 + 12 -10 + 6) \u003d 3 ^ (11). Answer: 3 ^ 11. In cases where various bases, the Rule A ^ N * B ^ n \u003d (A * B) ^ n works on the equal indicators. For example, 3 ^ 3 * 7 ^ 3 \u003d 21 ^ 3. Otherwise, when different bases and indicators, it is impossible to make full multiplication. Sometimes it is possible to partially simplify or resort to the help of computing technology.
The concept of degree in mathematics is introduced in the 7th grade in the classroom of algebra. And in the future, throughout the course of the study of mathematics, this concept is actively used in various types. The degree is a rather difficult topic, requiring storage of values \u200b\u200band skills correctly and quickly count. For faster and high-quality work with degrees of mathematics, the properties of the degree were invented. They help reduce large calculations, convert a huge example into one number to any extent. Properties are not so much, and they are all easily remembered and applied in practice. Therefore, the article discusses the basic properties of the degree, as well as where they are applied.
Properties of degree
We will look at the 12 properties of the degree, including the properties of degrees with the same bases, and for each property we give an example. Each of these properties will help you solve tasks with degrees, as well as save you from numerous computing errors.
1st property.
Many often forget about this property, make mistakes, representing a number in zero degree as zero.
2nd property.
3rd property.
It must be remembered that this property can only be applied when the numbers are performed, it does not work with the amount! And we must not forget that this is the following, properties apply only to degrees with the same bases.
4th property.
If a number is erected in the denominator to a negative degree, then when subtracting the degree of denominator is taken into the brackets to properly replace the sign at further computing.
The property works only during division, does not apply when subtracting!
5th property.
6th property.
This property can be applied in the opposite direction. The unit divided into some extent is the number in a minus degree.
7th property.
This property cannot be applied to sum and difference! When the amount or difference is erected, the formulas of abbreviated multiplication are used, and not the degree properties.
8th property.
9th property.
This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the root degree will vary depending on the denominator.
Also, this property is often used in reverse order. The root of any degree from among the number can be represented as the number to the degree unit divided by the degree of root. This property is very useful in cases if the root is not extracted.
10th property.
This property works not only with a square root and a second degree. If the degree of the root and the degree in which this root takes, they coincide, the answer will be the feeding expression.
11th property.
This property must be able to see in time when deciding to get rid of themselves from huge computing.
12th property.
Each of these properties will repeat to you in the tasks, it can be given in its pure form, and may require some transformations and the use of other formulas. Therefore, for the correct solution, only properties only know, you need to practice and connect other mathematical knowledge.
The use of degrees and their properties
They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, indicative equations and inequalities are solved, as well as degrees often complicate equations and examples related to other sections of mathematics. The degree helps to avoid large and long calculations, the degree is easier to reduce and calculate. But to work with large degrees, or with the degrees of large numbers, you need to know not only the degree properties, but to work correctly and with the grounds, be able to decompose them to facilitate the task. For convenience, the value of the numbers erected into a degree should be known. This will reduce your time when solving, eliminating the need for long computing.
The concept of degree plays a special role in logarithms. Since logarithm, in essence, is the degree of number.
Abbreviated multiplication formulas is another example of the use of degrees. They cannot be used by the properties of degrees, they are disclosed according to the special rules, but in each formula of abbreviated multiplication is invariably present.
The same degrees are actively used in physics and computer science. All transfers to the SI system are manufactured using degrees, and in the future, the properties of the degree are used in solving problems. The informatics are actively used decendible degrees, for the convenience of the account and simplify the perception of numbers. Further calculations for transfers of units of measurement or calculations of tasks, as well as in physics, occur using the degree properties.
Even degrees are very helpful in astronomy, it is rarely possible to apply the use of the properties of the degree, but the degree are actively used to reduce the recording of various quantities and distances.
Degrees are used in ordinary life, in calculating areas, volumes, distances.
With the help of degrees, it is written very large and very small values \u200b\u200bin any spheres of science.
Indicative equations and inequalities
A special place of the property of the degree occupies in the indicative equations and inequalities. These tasks are very often found, both in the school course and in the exams. All of them are solved through the use of the degree properties. The unknown is always in degree, so knowing all properties, it is not difficult to solve such an equation or inequality.
