With a natural indicator and its properties. Properties of degrees, formulations, proofs, examples
I. Work n factors, each of which is equal A called n-th power of the number A and is designated An.
Examples. Write the product as a degree.
1) mmmm; 2) aaabb; 3) 5 5 5 5 ccc; 4) ppkk+pppk-ppkkk.
Solution.
1) mmmm=m 4, since, by definition of a degree, the product of four factors, each of which is equal m, will fourth power of m.
2) aaabb=a 3 b 2 ; 3) 5·5·5·5·ccc=5 4 c 3 ; 4) ppkk+pppk-ppkkk=p 2 k 2 +p 3 k-p 2 k 3.
II. The action by which the product of several equal factors is found is called exponentiation. The number that is raised to a power is called the base of the power. The number that shows to what power the base is raised is called the exponent. So, An- degree, A– the basis of the degree, n– exponent. For example:
2 3 — it's a degree. Number 2 is the base of the degree, the exponent is equal to 3 . Degree value 2 3 equals 8, because 2 3 =2·2·2=8.
Examples. Write the following expressions without the exponent.
5) 4 3; 6) a 3 b 2 c 3 ; 7) a 3 -b 3 ; 8) 2a 4 +3b 2 .
Solution.
5) 4 3 = 4·4·4 ; 6) a 3 b 2 c 3 = aaabbccc; 7) a 3 -b 3 = aaa-bbb; 8) 2a 4 +3b 2 = 2aaaa+3bb.
III. and 0 =1 Any number (except zero) to the zero power is equal to one. For example, 25 0 =1.
IV. a 1 =aAny number to the first power is equal to itself.
V. a m∙ a n= a m + n When multiplying powers with on the same grounds the basis remains the same, and the indicators folded
Examples. Simplify:
9) a·a 3 ·a 7 ; 10) b 0 +b 2 b 3 ; 11) c 2 ·c 0 ·c·c 4 .
Solution.
9) a·a 3 ·a 7=a 1+3+7 =a 11 ; 10) b 0 +b 2 b 3 = 1+b 2+3 =1+b 5 ;
11) c 2 c 0 c c 4 = 1 c 2 c c 4 =c 2+1+4 =c 7 .
VI. a m: a n= a m - nWhen dividing powers with the same bases, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.
Examples. Simplify:
12) a 8:a 3 ; 13) m 11:m 4 ; 14) 5 6:5 4 .
12)a 8:a 3=a 8-3 =a 5 ; 13)m 11:m 4=m 11-4 =m 7; 14 ) 5 6:5 4 =5 2 =5·5=25.
VII. (a m) n= a mn When raising a power to a power, the base is left the same, and the exponents are multiplied.
Examples. Simplify:
15) (a 3) 4 ; 16) (c 5) 2.
15) (a 3) 4=a 3·4 =a 12 ; 16) (c 5) 2=c 5 2 =c 10.
note, which, since the product does not change from rearranging the factors, That:
15) (a 3) 4 = (a 4) 3 ; 16) (c 5) 2 = (c 2) 5 .
VI II. (a∙b) n =a n ∙b n When raising a product to a power, each of the factors is raised to that power.
Examples. Simplify:
17) (2a 2) 5 ; 18) 0.2 6 5 6 ; 19) 0.25 2 40 2.
Solution.
17) (2a 2) 5=2 5 ·a 2·5 =32a 10 ; 18) 0.2 6 5 6=(0.2·5) 6 =1 6 =1;
19) 0.25 2 40 2=(0.25·40) 2 =10 2 =100.
IX. When raising a fraction to a power, both the numerator and denominator of the fraction are raised to that power.
Examples. Simplify:
Solution.
Page 1 of 1 1
Lesson on the topic: "Rules of multiplication and division of powers with the same and different exponents. Examples"
Additional materials
Dear users, do not forget to leave your comments, reviews, wishes. All materials have been checked by an anti-virus program.
Teaching aids and simulators in the Integral online store for grade 7
Manual for the textbook Yu.N. Makarycheva Manual for the textbook by A.G. Mordkovich
Purpose of the lesson: learn to perform operations with powers of numbers.
First, let's remember the concept of "power of number". An expression of the form $\underbrace( a * a * \ldots * a )_(n)$ can be represented as $a^n$.
The converse is also true: $a^n= \underbrace( a * a * \ldots * a )_(n)$.
This equality is called “recording the degree as a product.” It will help us determine how to multiply and divide powers.
Remember:
a– the basis of the degree.
n– exponent.
If n=1, which means the number A took once and accordingly: $a^n= 1$.
If n= 0, then $a^0= 1$.
We can find out why this happens when we get acquainted with the rules of multiplication and division of powers.
Multiplication rules
a) If powers with the same base are multiplied.To get $a^n * a^m$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( a * a * \ldots * a )_(m )$.
The figure shows that the number A have taken n+m times, then $a^n * a^m = a^(n + m)$.
Example.
$2^3 * 2^2 = 2^5 = 32$.
This property is convenient to use to simplify the work when raising a number to a higher power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.
b) If degrees with different bases, but the same exponent are multiplied.
To get $a^n * b^n$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( b * b * \ldots * b )_(m )$.
If we swap the factors and count the resulting pairs, we get: $\underbrace( (a * b) * (a * b) * \ldots * (a * b) )_(n)$.
So $a^n * b^n= (a * b)^n$.
Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.
Division rules
a) The basis of the degree is the same, the indicators are different.Consider dividing a power with a larger exponent by dividing a power with a smaller exponent.
So, we need $\frac(a^n)(a^m)$, Where n>m.
Let's write the degrees as a fraction:
$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( a * a * \ldots * a )_(m))$.
For convenience, we write the division as a simple fraction.Now let's reduce the fraction.
It turns out: $\underbrace( a * a * \ldots * a )_(n-m)= a^(n-m)$.
Means, $\frac(a^n)(a^m)=a^(n-m)$.
This property will help explain the situation with raising a number to the zero power. Let's assume that n=m, then $a^0= a^(n-n)=\frac(a^n)(a^n) =1$.
Examples.
$\frac(3^3)(3^2)=3^(3-2)=3^1=3$.
$\frac(2^2)(2^2)=2^(2-2)=2^0=1$.
b) The bases of the degree are different, the indicators are the same.
Let's say we need $\frac(a^n)( b^n)$. Let's write powers of numbers as fractions:
$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( b * b * \ldots * b )_(n))$.
For convenience, let's imagine.
Using the property of fractions, we divide the large fraction into the product of small ones, we get.
$\underbrace( \frac(a)(b) * \frac(a)(b) * \ldots * \frac(a)(b) )_(n)$.
Accordingly: $\frac(a^n)( b^n)=(\frac(a)(b))^n$.
Example.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.
primary goal
To familiarize students with the properties of degrees with natural exponents and teach them how to perform operations with degrees.
Topic “Degree and its properties” includes three questions:
- Determination of degree with a natural indicator.
- Multiplication and division of powers.
- Exponentiation of product and degree.
Control questions
- Formulate the definition of a degree with a natural exponent greater than 1. Give an example.
- Formulate the definition of degree with exponent 1. Give an example.
- What is the order of operations when calculating the value of an expression containing powers?
- Formulate the main property of degree. Give an example.
- Formulate the rule for multiplying powers with the same bases. Give an example.
- Formulate a rule for dividing powers with the same bases. Give an example.
- Formulate the rule for exponentiation of a product. Give an example. Prove the identity (ab) n = a n b n .
- Formulate the rule for raising a power to a power. Give an example. Prove the identity (a m) n = a m n .
Definition of degree.
Power of number a with natural indicator n, greater than 1, is the product of n factors, each of which is equal A. Power of number A with exponent 1 is the number itself A.
Degree with base A and indicator n is written like this: and n. It reads “ A to a degree n”; “ nth power of a number A ”.
By definition of degree:
a 4 = a a a a
. . . . . . . . . . . .
Finding the value of a power is called by exponentiation .
1. Examples of exponentiation:
3 3 = 3 3 3 = 27
0 4 = 0 0 0 0 = 0
(-5) 3 = (-5) (-5) (-5) = -125
25 ; 0,09 ;
25 = 5 2 ; 0,09 = (0,3) 2 ; .
27 ; 0,001 ; 8 .
27 = 3 3 ; 0,001 = (0,1) 3 ; 8 = 2 3 .
4. Find the meanings of the expressions:
a) 3 10 3 = 3 10 10 10 = 3 1000 = 3000
b) -2 4 + (-3) 2 = 7
2 4 = 16
(-3) 2 = 9
-16 + 9 = 7
Option 1
a) 0.3 0.3 0.3
c) b b b b b b b
d) (-x) (-x) (-x) (-x)
e) (ab) (ab) (ab)
2. Present the number as a square:
3. Present the numbers as a cube:
4. Find the meanings of the expressions:
c) -1 4 + (-2) 3
d) -4 3 + (-3) 2
e) 100 - 5 2 4
Multiplication of powers.
For any number a and arbitrary numbers m and n the following holds:
a m a n = a m + n .
Proof:
Rule : When multiplying powers with the same bases, the bases are left the same, and the exponents of the powers are added.
a m a n a k = a m + n a k = a (m + n) + k = a m + n + k
a) x 5 x 4 = x 5 + 4 = x 9
b) y y 6 = y 1 y 6 = y 1 + 6 = y 7
c) b 2 b 5 b 4 = b 2 + 5 + 4 = b 11
d) 3 4 9 = 3 4 3 2 = 3 6
e) 0.01 0.1 3 = 0.1 2 0.1 3 = 0.1 5
a) 2 3 2 = 2 4 = 16
b) 3 2 3 5 = 3 7 = 2187
Option 1
1. Present as a degree:
a) x 3 x 4 e) x 2 x 3 x 4
b) a 6 a 2 g) 3 3 9
c) y 4 y h) 7 4 49
d) a a 8 i) 16 2 7
e) 2 3 2 4 j) 0.3 3 0.09
2. Present as a degree and find the value from the table:
a) 2 2 2 3 c) 8 2 5
b) 3 4 3 2 d) 27 243
Division of degrees.
For any number a0 and arbitrary natural numbers m and n, such that m>n the following holds:
a m: a n = a m - n
Proof:
a m - n a n = a (m - n) + n = a m - n + n = a m
by definition of quotient:
a m: a n = a m - n .
Rule: When dividing powers with the same bases, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.
Definition: The power of a number a, not equal to zero, with a zero exponent is equal to one:
because a n: a n = 1 at a0.
a) x 4: x 2 = x 4 - 2 = x 2
b) y 8: y 3 = y 8 - 3 = y 5
c) a 7:a = a 7:a 1 = a 7 - 1 = a 6
d) from 5:from 0 = from 5:1 = from 5
a) 5 7:5 5 = 5 2 = 25
b) 10 20:10 17 = 10 3 = 1000
V)
G)
d)
Option 1
1. Present the quotient as a power:
2. Find the meanings of the expressions:
Raising to the power of a product.
For any a and b and an arbitrary natural number n:
(ab) n = a n b n
Proof:
By definition of degree
(ab)n= 
Grouping separately the factors a and the factors b, we get:
= 
The proven property of the power of a product extends to the power of the product of three or more factors.
For example:
(a b c) n = a n b n c n ;
(a b c d) n = a n b n c n d n .
Rule: When raising a product to a power, each factor is raised to that power and the result is multiplied.
1. Raise to a power:
a) (a b) 4 = a 4 b 4
b) (2 x y) 3 =2 3 x 3 y 3 = 8 x 3 y 3
c) (3 a) 4 = 3 4 a 4 = 81 a 4
d) (-5 y) 3 = (-5) 3 y 3 = -125 y 3
e) (-0.2 x y) 2 = (-0.2) 2 x 2 y 2 = 0.04 x 2 y 2
e) (-3 a b c) 4 = (-3) 4 a 4 b 4 c 4 = 81 a 4 b 4 c 4
2. Find the value of the expression:
a) (2 10) 4 = 2 4 10 4 = 16 1000 = 16000
b) (3 5 20) 2 = 3 2 100 2 = 9 10000= 90000
c) 2 4 5 4 = (2 5) 4 = 10 4 = 10000
d) 0.25 11 4 11 = (0.25 4) 11 = 1 11 = 1
d)
Option 1
1. Raise to a power:
b) (2 a c) 4
e) (-0.1 x y) 3
2. Find the value of the expression:
b) (5 7 20) 2
Raising to a power of a power.
For any number a and arbitrary natural numbers m and n:
(a m) n = a m n
Proof:
By definition of degree
(a m) n = 
Rule: When raising a power to a power, the base is left the same, and the exponents are multiplied.
1. Raise to a power:
(a 3) 2 = a 6 (x 5) 4 = x 20
(y 5) 2 = y 10 (b 3) 3 = b 9
2. Simplify the expressions:
a) a 3 (a 2) 5 = a 3 a 10 = a 13
b) (b 3) 2 b 7 = b 6 b 7 = b 13
c) (x 3) 2 (x 2) 4 = x 6 x 8 = x 14
d) (y 7) 3 = (y 8) 3 = y 24
A)
b)
Option 1
1. Raise to a power:
a) (a 4) 2 b) (x 4) 5
c) (y 3) 2 d) (b 4) 4
2. Simplify the expressions:
a) a 4 (a 3) 2
b) (b 4) 3 b 5+
c) (x 2) 4 (x 4) 3
d) (y 9) 2
3. Find the meaning of the expressions:
Application
Definition of degree.
Option 2
1st Write the product as a power:
a) 0.4 0.4 0.4
c) a a a a a a a a
d) (-y) (-y) (-y) (-y)
e) (bс) (bс) (bс)
2. Present the number as a square:
3. Present the numbers as a cube:
4. Find the meanings of the expressions:
c) -1 3 + (-2) 4
d) -6 2 + (-3) 2
e) 4 5 2 – 100
Option 3
1. Write the product as a power:
a) 0.5 0.5 0.5
c) with with with with with with with with with
d) (-x) (-x) (-x) (-x)
e) (ab) (ab) (ab)
2. Present the number as a square: 100; 0.49; .
3. Present the numbers as a cube:
4. Find the meanings of the expressions:
c) -1 5 + (-3) 2
d) -5 3 + (-4) 2
e) 5 4 2 - 100
Option 4
1. Write the product as a power:
a) 0.7 0.7 0.7
c) x x x x x x
d) (-a) (-a) (-a)
e) (bс) (bс) (bс) (bc)
2. Present the number as a square:
3. Present the numbers as a cube:
4. Find the meanings of the expressions:
c) -1 4 + (-3) 3
d) -3 4 + (-5) 2
e) 100 - 3 2 5
Multiplication of powers.
Option 2
1. Present as a degree:
a) x 4 x 5 e) x 3 x 4 x 5
b) a 7 a 3 g) 2 3 4
c) y 5 y h) 4 3 16
d) a a 7 i) 4 2 5
e) 2 2 2 5 j) 0.2 3 0.04
2. Present as a degree and find the value from the table:
a) 3 2 3 3 c) 16 2 3
b) 2 4 2 5 d) 9 81
Option 3
1. Present as a degree:
a) a 3 a 5 f) y 2 y 4 y 6
b) x 4 x 7 g) 3 5 9
c) b 6 b h) 5 3 25
d) y 8 i) 49 7 4
e) 2 3 2 6 j) 0.3 4 0.27
2. Present as a degree and find the value from the table:
a) 3 3 3 4 c) 27 3 4
b) 2 4 2 6 d) 16 64
Option 4
1. Present as a degree:
a) a 6 a 2 e) x 4 x x 6
b) x 7 x 8 g) 3 4 27
c) y 6 y h) 4 3 16
d) x x 10 i) 36 6 3
e) 2 4 2 5 j) 0.2 2 0.008
2. Present as a degree and find the value from the table:
a) 2 6 2 3 c) 64 2 4
b) 3 5 3 2 d) 81 27
Division of degrees.
Option 2
1. Present the quotient as a power:
2. Find the meanings of the expressions.
Video tutorial 2: Degree with a natural indicator and its properties
Lecture:
Degree with natural indicator
Under degree some number "A" with some indicator "n" understand the product of a number "A" on its own "n" once.
When we talk about a degree with a natural exponent, it means that the number "n" must be integer and not negative.
A- the base of the degree, which shows which number should be multiplied by itself,
n- exponent - it tells how many times the base needs to be multiplied by itself.
For example:
8 4 = 8 * 8 * 8 * 8 = 4096.
In this case, the base of the degree is understood to be the number “8”, the exponent of the degree is the number “4”, and the value of the degree is the number “4096”.
The biggest and most common mistake when calculating a degree is multiplying the exponent by the base - THIS IS NOT CORRECT!
When we talk about a degree with a natural exponent, we mean that only the exponent (n) must be a natural number.
You can take any number on the number line as a base.
For example,
(-0,1) 3 = (-0,1) * (-0,1) * (-0,1) = (-0,001).
The mathematical operation that is performed on the base and exponent is called exponentiation.
Addition\subtraction is a mathematical operation of the first stage, multiplication\division is an action of the second stage, raising a power is a mathematical action of the third stage, that is, one of the highest.
This hierarchy of mathematical operations determines the order in the calculation. If this action occurs in tasks among the previous two, then it is done first.
For example:
15 + 6 *2 2 = 39
In this example, you must first raise 2 to the power, that is,
then multiply the result by 6, that is
The power with a natural exponent is used not only for specific calculations, but also for the convenience of writing large numbers. In this case, the concept is also used "standard form of number". This notation involves multiplying a certain number from 1 to 9 by a power equal to 10 with some exponent.
For example, to record the radius of the Earth in standard form, use the following notation:
6400000 m = 6.4 * 10 6 m,
and the mass of the Earth, for example, is written as follows:
Properties of degree
For the convenience of solving examples with degrees, you need to know their basic properties:
1. If you need to multiply two powers that have the same base, then in this case the base must be left unchanged and the exponents added.
a n * a m = a n+m
For example:
5 2 * 5 4 = 5 6 .
2. If it is necessary to divide two degrees that have the same bases, then in this case the base must be left unchanged and the exponents subtracted. Please note that for operations with powers with a natural exponent, the exponent of the dividend must be greater than the exponent of the divisor. Otherwise, the quotient of this action will be a number with a negative exponent.
a n / a m = a n-m
For example,
5 4 * 5 2 = 5 2 .
3. If it is necessary to raise one power to another, the same number remains the base of the result, and the exponents are multiplied.
(a n) m = a n*m
For example,
4. If it is necessary to raise the product of arbitrary numbers to a certain power, then you can use a certain distributive law, under which we obtain the product of different bases to the same power.
(a * b) m = a m * b m
For example,
(5 * 8) 2 = 5 2 * 8 2 .
5. A similar property can be used to divide powers, in other words, to raise an ordinary double to a power.
(a / b) m = a m / b m
6. Any number that is raised to an exponent equal to one is equal to the original number.
a 1 = a
For example,
7. When raising any number to a power with exponent zero, the result of this calculation will always be one.
and 0 = 1
For example,
| | |
Earlier we already talked about what a power of a number is. It has certain properties that are useful in solving problems: these are all possible indicators degrees we will examine in this article. We will also clearly show with examples how they can be proven and correctly applied in practice.
Yandex.RTB R-A-339285-1
Let us recall the previously formulated concept of a degree with a natural exponent: this is the product of the nth number of factors, each of which is equal to a. We will also need to remember how to multiply real numbers correctly. All this will help us formulate the following properties for a degree with a natural exponent:
Definition 1
1. The main property of the degree: a m · a n = a m + n
Can be generalized to: a n 1 · a n 2 · … · a n k = a n 1 + n 2 + … + n k .
2. Property of the quotient for degrees having the same bases: a m: a n = a m − n
3. Product degree property: (a · b) n = a n · b n
The equality can be expanded to: (a 1 · a 2 · … · a k) n = a 1 n · a 2 n · … · a k n
4. Property of quotient to natural degree: (a: b) n = a n: b n
5. Raise the power to the power: (a m) n = a m n ,
Can be generalized to: (((a n 1) n 2) …) n k = a n 1 · n 2 · … · n k
6. Compare the degree with zero:
- if a > 0, then for any natural number n, a n will be greater than zero;
- with a equal to 0, a n will also be equal to zero;
- at a< 0 и таком показателе степени, который будет четным числом 2 · m , a 2 · m будет больше нуля;
- at a< 0 и таком показателе степени, который будет нечетным числом 2 · m − 1 , a 2 · m − 1 будет меньше нуля.
7. Equality a n< b n будет справедливо для любого натурального n при условии, что a и b больше нуля и не равны друг другу.
8. The inequality a m > a n will be true provided that m and n are natural numbers, m is greater than n and a is greater than zero and less than one.
As a result, we got several equalities; if all the conditions stated above are met, they will be identical. For each of the equalities, for example, for the main property, you can swap the right and left side: a m · a n = a m + n - the same as a m + n = a m · a n. In this form it is often used to simplify expressions.
1. Let's start with the basic property of degree: the equality a m · a n = a m + n will be true for any natural m and n and real a. How to prove this statement?
The basic definition of powers with natural exponents will allow us to transform equality into a product of factors. We will get a record like this:
This can be shortened to 
(remember the basic properties of multiplication). As a result, we got the power of the number a with natural exponent m + n. Thus, a m + n, which means the main property of the degree has been proven.
Let's look at a specific example that confirms this.
Example 1
So we have two powers with base 2. Their natural indicators are 2 and 3, respectively. We have the equality: 2 2 · 2 3 = 2 2 + 3 = 2 5 Let's calculate the values to check the validity of this equality.
Let's perform the necessary mathematical operations: 2 2 2 3 = (2 2) (2 2 2) = 4 8 = 32 and 2 5 = 2 2 2 2 2 = 32
As a result, we got: 2 2 · 2 3 = 2 5. The property has been proven.
Due to the properties of multiplication, we can generalize the property by formulating it in the form of three and more powers whose exponents are natural numbers and whose bases are the same. If we denote the number of natural numbers n 1, n 2, etc. by the letter k, we get the correct equality:
a n 1 · a n 2 · … · a n k = a n 1 + n 2 + … + n k .
Example 2
2. Next, we need to prove the following property, which is called the quotient property and is inherent in powers with the same bases: this is the equality a m: a n = a m − n, which is valid for any natural m and n (and m is greater than n)) and any non-zero real a .
To begin with, let us clarify what exactly is the meaning of the conditions that are mentioned in the formulation. If we take a equal to zero, then we end up with division by zero, which we cannot do (after all, 0 n = 0). The condition that the number m must be greater than n is necessary so that we can stay within the limits of natural exponents: subtracting n from m, we get natural number. If the condition is not met, we will end up with a negative number or zero, and again we will go beyond the study of degrees with natural exponents.
Now we can move on to the proof. From what we have previously studied, let us recall the basic properties of fractions and formulate the equality as follows:
a m − n · a n = a (m − n) + n = a m
From it we can deduce: a m − n · a n = a m
Let's remember the connection between division and multiplication. It follows from it that a m − n is the quotient of the powers a m and a n . This is the proof of the second property of degree.
Example 3
For clarity, let’s substitute specific numbers into the exponents, and denote the base of the degree as π : π 5: π 2 = π 5 − 3 = π 3
3. Next we will analyze the property of the power of a product: (a · b) n = a n · b n for any real a and b and natural n.
According to the basic definition of a power with a natural exponent, we can reformulate the equality as follows:
Recalling the properties of multiplication, we write: 
. This means the same as a n · b n .
Example 4
2 3 · - 4 2 5 4 = 2 3 4 · - 4 2 5 4
If we have three or more factors, then this property also applies to this case. Let us introduce the notation k for the number of factors and write:
(a 1 · a 2 · … · a k) n = a 1 n · a 2 n · … · a k n
Example 5
With specific numbers we get the following correct equality: (2 · (- 2 , 3) · a) 7 = 2 7 · (- 2 , 3) 7 · a
4. After this, we will try to prove the property of the quotient: (a: b) n = a n: b n for any real a and b, if b is not equal to 0 and n is a natural number.
To prove this, you can use the previous property of degrees. If (a: b) n · b n = ((a: b) · b) n = a n , and (a: b) n · b n = a n , then it follows that (a: b) n is the quotient of dividing a n by b n.
Example 6
Let's calculate an example: 3 1 2: - 0. 5 3 = 3 1 2 3: (- 0 , 5) 3
Example 7
Let's start right away with an example: (5 2) 3 = 5 2 3 = 5 6
Now let’s formulate a chain of equalities that will prove to us that the equality is true: 
If we have degrees of degrees in the example, then this property is also true for them. If we have any natural numbers p, q, r, s, then it will be true:
a p q y s = a p q y s
Example 8
Let's add some specifics: (((5 , 2) 3) 2) 5 = (5 , 2) 3 + 2 + 5 = (5 , 2) 10
6. Another property of powers with a natural exponent that we need to prove is the property of comparison.
First, let's compare the degree to zero. Why does a n > 0, provided that a is greater than 0?
If we multiply one positive number by another, we also get a positive number. Knowing this fact, we can say that it does not depend on the number of factors - the result of multiplying any number of positive numbers is a positive number. What is a degree if not the result of multiplying numbers? Then for any power a n with a positive base and natural exponent this will be true.
Example 9
3 5 > 0 , (0 , 00201) 2 > 0 and 34 9 13 51 > 0
It is also obvious that a power with a base equal to zero is itself zero. No matter what power we raise zero to, it will remain zero.
Example 10
0 3 = 0 and 0 762 = 0
If the base of the degree is a negative number, then the proof is a little more complicated, since the concept of even/odd exponent becomes important. Let us first take the case when the exponent is even, and denote it 2 · m, where m is a natural number.
Let's remember how to multiply correctly negative numbers: the product a · a is equal to the product of the moduli, and therefore it will be a positive number. Then 
and the degree a 2 m are also positive.
Example 11
For example, (− 6) 4 > 0, (− 2, 2) 12 > 0 and - 2 9 6 > 0
What if the exponent with a negative base is an odd number? Let's denote it 2 · m − 1 .
Then 
All products a · a, according to the properties of multiplication, are positive, and so is their product. But if we multiply it by the only remaining number a, then the final result will be negative.
Then we get: (− 5) 3< 0 , (− 0 , 003) 17 < 0 и - 1 1 102 9 < 0
How to prove this?
a n< b n – неравенство, представляющее собой произведение левых и правых частей nверных неравенств a < b . Вспомним основные свойства неравенств справедливо и a n < b n .
Example 12
For example, the following inequalities are true: 3 7< (2 , 2) 7 и 3 5 11 124 > (0 , 75) 124
8. We just have to prove the last property: if we have two powers whose bases are identical and positive, and whose exponents are natural numbers, then the one whose exponent is smaller is greater; and of two powers with natural exponents and identical bases greater than one, the one whose exponent is greater is greater.
Let us prove these statements.
First we need to make sure that a m< a n при условии, что m больше, чем n , и а больше 0 , но меньше 1 .Теперь сравним с нулем разность a m − a n
Let's take a n out of brackets, after which our difference will take the form a n · (a m − n − 1) . Its result will be negative (because the result of multiplying a positive number by a negative number is negative). After all, according to the initial conditions, m − n > 0, then a m − n − 1 is negative, and the first factor is positive, like any natural degree with a positive basis.
It turned out that a m − a n< 0 и a m < a n . Свойство доказано.
It remains to prove the second part of the statement formulated above: a m > a is true for m > n and a > 1. Let us indicate the difference and put a n out of brackets: (a m − n − 1). The power of a n for a greater than one will give positive result; and the difference itself will also turn out to be positive due to the initial conditions, and for a > 1 the degree a m − n is greater than one. It turns out that a m − a n > 0 and a m > a n , which is what we needed to prove.
Example 13
Example with specific numbers: 3 7 > 3 2
Basic properties of degrees with integer exponents
For powers with positive integer exponents, the properties will be similar, because positive integers are natural numbers, which means that all the equalities proved above are also true for them. They are also suitable for cases where the exponents are negative or equal to zero (provided that the base of the degree itself is non-zero).
Thus, the properties of powers are the same for any bases a and b (provided that these numbers are real and not equal to 0) and any exponents m and n (provided that they are integers). Let us write them briefly in the form of formulas:
Definition 2
1. a m · a n = a m + n
2. a m: a n = a m − n
3. (a · b) n = a n · b n
4. (a: b) n = a n: b n
5. (a m) n = a m n
6. a n< b n и a − n >b − n subject to positive integer n, positive a and b, a< b
7.am< a n , при условии целых m и n , m >n and 0< a < 1 , при a >1 a m > a n .
If the base of the degree is zero, then the entries a m and a n make sense only in the case of natural and positive m and n. As a result, we find that the formulations above are also suitable for cases with a power with a zero base, if all other conditions are met.
The proofs of these properties in this case are simple. We will need to remember what a degree with a natural and integer exponent is, as well as the properties of operations with real numbers.
Let's look at the power-to-power property and prove that it is true for both positive and non-positive integers. Let's start by proving the equalities (a p) q = a p · q, (a − p) q = a (− p) · q, (a p) − q = a p · (− q) and (a − p) − q = a (− p) · (− q)
Conditions: p = 0 or natural number; q – similar.
If the values of p and q are greater than 0, then we get (a p) q = a p · q. We have already proved a similar equality before. If p = 0, then:
(a 0) q = 1 q = 1 a 0 q = a 0 = 1
Therefore, (a 0) q = a 0 q
For q = 0 everything is exactly the same:
(a p) 0 = 1 a p 0 = a 0 = 1
Result: (a p) 0 = a p · 0 .
If both indicators are zero, then (a 0) 0 = 1 0 = 1 and a 0 · 0 = a 0 = 1, which means (a 0) 0 = a 0 · 0.
Let us recall the property of quotients to a degree proved above and write:
1 a p q = 1 q a p q
If 1 p = 1 1 … 1 = 1 and a p q = a p q, then 1 q a p q = 1 a p q
We can transform this notation by virtue of the basic rules of multiplication into a (− p) · q.
Also: a p - q = 1 (a p) q = 1 a p · q = a - (p · q) = a p · (- q) .
And (a - p) - q = 1 a p - q = (a p) q = a p q = a (- p) (- q)
The remaining properties of the degree can be proved in a similar way by transforming the existing inequalities. We will not dwell on this in detail; we will only point out the difficult points.
Proof of the penultimate property: remember, a − n > b − n is true for any integers negative values nand any positive a and b, provided that a is less than b.
Then the inequality can be transformed as follows:
1 a n > 1 b n
Let's write the right and left sides as a difference and perform the necessary transformations:
1 a n - 1 b n = b n - a n a n · b n
Recall that in the condition a is less than b, then, according to the definition of a degree with a natural exponent: - a n< b n , в итоге: b n − a n > 0 .
a n · b n ends up being a positive number because its factors are positive. As a result, we have the fraction b n - a n a n · b n, which ultimately also gives a positive result. Hence 1 a n > 1 b n whence a − n > b − n , which is what we needed to prove.
The last property of powers with integer exponents is proven similarly to the property of powers with natural exponents.
Basic properties of powers with rational exponents
In previous articles, we looked at what a degree with a rational (fractional) exponent is. Their properties are the same as those of degrees with integer exponents. Let's write down:
Definition 3
1. a m 1 n 1 · a m 2 n 2 = a m 1 n 1 + m 2 n 2 for a > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 (product property degrees with the same bases).
2. a m 1 n 1: b m 2 n 2 = a m 1 n 1 - m 2 n 2, if a > 0 (quotient property).
3. a · b m n = a m n · b m n for a > 0 and b > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 and (or) b ≥ 0 (product property in fractional degree).
4. a: b m n = a m n: b m n for a > 0 and b > 0, and if m n > 0, then for a ≥ 0 and b > 0 (the property of a quotient to a fractional power).
5. a m 1 n 1 m 2 n 2 = a m 1 n 1 · m 2 n 2 for a > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 (property of degree in degrees).
6.a p< b p при условии любых положительных a и b , a < b и рациональном p при p >0 ; if p< 0 - a p >b p (the property of comparing powers with equal rational exponents).
7.a p< a q при условии рациональных чисел p и q , p >q at 0< a < 1 ; если a >0 – a p > a q
To prove these provisions, we need to remember what a degree with a fractional exponent is, what are the properties of the arithmetic root of the nth degree, and what are the properties of a degree with integer exponents. Let's look at each property.
According to what a degree with a fractional exponent is, we get:
a m 1 n 1 = a m 1 n 1 and a m 2 n 2 = a m 2 n 2, therefore, a m 1 n 1 · a m 2 n 2 = a m 1 n 1 · a m 2 n 2
The properties of the root will allow us to derive equalities:
a m 1 m 2 n 1 n 2 a m 2 m 1 n 2 n 1 = a m 1 n 2 a m 2 n 1 n 1 n 2
From this we get: a m 1 · n 2 · a m 2 · n 1 n 1 · n 2 = a m 1 · n 2 + m 2 · n 1 n 1 · n 2
Let's convert:
a m 1 · n 2 · a m 2 · n 1 n 1 · n 2 = a m 1 · n 2 + m 2 · n 1 n 1 · n 2
The exponent can be written as:
m 1 n 2 + m 2 n 1 n 1 n 2 = m 1 n 2 n 1 n 2 + m 2 n 1 n 1 n 2 = m 1 n 1 + m 2 n 2
This is the proof. The second property is proven in exactly the same way. Let's write a chain of equalities:
a m 1 n 1: a m 2 n 2 = a m 1 n 1: a m 2 n 2 = a m 1 n 2: a m 2 n 1 n 1 n 2 = = a m 1 n 2 - m 2 n 1 n 1 n 2 = a m 1 n 2 - m 2 n 1 n 1 n 2 = a m 1 n 2 n 1 n 2 - m 2 n 1 n 1 n 2 = a m 1 n 1 - m 2 n 2
Proofs of the remaining equalities:
a · b m n = (a · b) m n = a m · b m n = a m n · b m n = a m n · b m n ; (a: b) m n = (a: b) m n = a m: b m n = = a m n: b m n = a m n: b m n ; a m 1 n 1 m 2 n 2 = a m 1 n 1 m 2 n 2 = a m 1 n 1 m 2 n 2 = = a m 1 m 2 n 1 n 2 = a m 1 m 2 n 1 n 2 = = a m 1 m 2 n 2 n 1 = a m 1 m 2 n 2 n 1 = a m 1 n 1 m 2 n 2
Next property: let us prove that for any values of a and b greater than 0, if a is less than b, a p will be satisfied< b p , а для p больше 0 - a p >b p
Let's represent the rational number p as m n. In this case, m is an integer, n is a natural number. Then conditions p< 0 и p >0 will extend to m< 0 и m >0 . For m > 0 and a< b имеем (согласно свойству степени с целым положительным показателем), что должно выполняться неравенство a m < b m .
We use the property of roots and output: a m n< b m n
Taking into account the positive values of a and b, we rewrite the inequality as a m n< b m n . Оно эквивалентно a p < b p .
In the same way for m< 0 имеем a a m >b m , we get a m n > b m n which means a m n > b m n and a p > b p .
It remains for us to provide a proof of the last property. Let us prove that for rational numbers p and q, p > q at 0< a < 1 a p < a q , а при a >0 will be true a p > a q .
Rational numbers p and q can be reduced to a common denominator and get the fractions m 1 n and m 2 n
Here m 1 and m 2 are integers, and n is a natural number. If p > q, then m 1 > m 2 (taking into account the rule for comparing fractions). Then at 0< a < 1 будет верно a m 1 < a m 2 , а при a >1 – inequality a 1 m > a 2 m.
They can be rewritten as follows:
a m 1 n< a m 2 n a m 1 n >a m 2 n
Then you can make transformations and end up with:
a m 1 n< a m 2 n a m 1 n >a m 2 n
To summarize: for p > q and 0< a < 1 верно a p < a q , а при a >0 – a p > a q .
Basic properties of powers with irrational exponents
To such a degree one can extend all the properties described above that a degree with rational exponents has. This follows from its very definition, which we gave in one of the previous articles. Let us briefly formulate these properties (conditions: a > 0, b > 0, exponents p and q are irrational numbers):
Definition 4
1. a p · a q = a p + q
2. a p: a q = a p − q
3. (a · b) p = a p · b p
4. (a: b) p = a p: b p
5. (a p) q = a p · q
6.a p< b p верно при любых положительных a и b , если a < b и p – иррациональное число больше 0 ; если p меньше 0 , то a p >b p
7.a p< a q верно, если p и q – иррациональные числа, p < q , 0 < a < 1 ; если a >0, then a p > a q.
Thus, all powers whose exponents p and q are real numbers, provided a > 0, have the same properties.
If you notice an error in the text, please highlight it and press Ctrl+Enter