If two powers are multiplied (or divided), which have different bases, but the same exponents, then their bases can be multiplied (or divided), and the exponent of the result can be left the same as that of the factors (or dividend and divisor).
In general, in mathematical language, these rules are written as follows:
a m × b m = (ab) m
a m ÷ b m = (a/b) m
When dividing, b cannot be equal to 0, that is, the second rule must be supplemented with the condition b ≠ 0.
Examples:
2 3 × 3 3 = (2 × 3) 3 = 63 = 36 × 6 = 180 + 36 = 216
6 5 ÷ 3 5 = (6 ÷ 3) 5 = 2 5 = 32
Now, using these specific examples, we will prove that the rules-properties of degrees with the same exponents are correct. Let's solve these examples as if we don't know about the properties of degrees:
2 3 × 3 3 = (2 × 2 × 2) × (3 × 3 × 3) = 2 × 2 × 2 × 3 × 3 × 3 = 8 × 27 = 160 + 56 = 216
65 ÷ 35 = (6 × 6 × 6 × 6 × 6) ÷ (3 × 3 × 3 × 3 × 3) == 2 × 2 × 2 × 2 × 2 = 32
As we can see, the answers coincided with those obtained when the rules were used. Knowing these rules allows you to simplify calculations.
Note that the expression 2 × 2 × 2 × 3 × 3 × 3 can be written as follows:
(2 × 3) × (2 × 3) × (2 × 3).
This expression in turn is something other than (2 × 3) 3. that is, 6 3.
The considered properties of degrees with the same indicators can be used in the opposite direction. For example, what is 18 2?
18 2 = (3 × 3 × 2) 2 = 3 2 × 3 2 × 2 2 = 9 × 9 × 4 = 81 × 4 = 320 + 4 = 324
Properties of powers are also used when solving examples:
= 2 4 × 3 6 = 2 4 × 3 4 × 3 × 3 = 6 4 × 3 2 = 6 2 × 6 2 × 3 2 = (6 × 6 × 3) 2 = 108 2 = 108 × 108 = 108 ( 100 + 8) = 10800 + 864 = 11664
Basic properties of degrees
"Properties of degrees" is a fairly popular query in search engines, which shows great interest in the properties of the degree. We have collected for you all the properties of a degree (properties of a degree with a natural exponent, properties of a degree with a rational exponent, properties of a degree with an integer exponent) in one place. You can download a short version of the cheat sheet "Properties of degrees" in .pdf format so that, if necessary, you can easily remember them or familiarize yourself with them properties of degrees directly on the site. In details properties of powers with examples discussed below.
Download the cheat sheet "Properties of degrees" (format.pdf)
Properties of degrees (briefly)
a 0=1 if a≠0
a 1=a
(−a)n=an, If n- even
(−a)n=−an, If n- odd
(a⋅b)n=an⋅bn
(ab)n=anbn
a−n=1an
(ab)−n=(ba)n
an⋅am=an+m
anam=an−m
(an)m=an⋅m
Properties of degrees (with examples)
1st degree property Any number other than zero to the zero power is equal to one. a 0=1 if a≠0 For example: 1120=1, (−4)0=1, (0,15)0=1
2nd degree property Any number to the first power is equal to the number itself. a 1=a For example: 231=23, (−9,3)1=−9,3
3rd degree property Any number to an even power is positive. an=an, If n- even (divisible by 2) integer (− a)n=an, If n- even (divisible by 2) integer For example: 24=16, (−3)2=32=9, (−1)10=110=1
4th degree property Any number to an odd power retains its sign. an=an, If n- odd (not divisible by 2) integer (− a)n=−an, If n- odd (not divisible by 2) integer For example: 53=125, (−3)3=33=27, (−1)11=−111=−1
5th degree property Product of numbers raised oh to a power, can be represented as the product of numbers raised s V this degree (and vice versa). ( a⋅b)n=an⋅bn, wherein a, b, n For example: (2,1⋅0,3)4,5=2,14,5⋅0,34,5
6th degree property The quotient (division) of numbers raised oh to a power, can be represented as the quotient of numbers raised s V this degree (and vice versa). ( ab)n=anbn, wherein a, b, n- any valid (not necessarily integer) numbers For example: (1,75)0,1=(1,7)0,150,1
7th degree property Any number to a negative power is equal to its reciprocal number to that power. (The reciprocal is the number by which the given number must be multiplied to get one.) a−n=1an, wherein a And n- any valid (not necessarily integer) numbers For example: 7−2=172=149
8th degree property Any fraction to a negative power is equal to the reciprocal fraction to that power. ( ab)−n=(ba)n, wherein a, b, n- any valid (not necessarily integer) numbers For example: (23)−2=(32)2, (14)−3=(41)3=43=64
9th degree property When multiplying powers with the same base, the exponents are added, but the base remains the same. an⋅am=an+m, wherein a, n, m- any valid (not necessarily integer) numbers For example: 23⋅25=23+5=28, note that this property of the degree is preserved for negative values of the degrees 3−2⋅36=3−2+6=34, 47⋅4−3=47+(−3)= 47−3=44
10th degree property When dividing powers with the same base, the exponents are subtracted, but the base remains the same. anam=an−m, wherein a, n, m- any valid (not necessarily integer) numbers For example:(1,4)2(1,4)3=1.42+3=1.45, note how this power property applies to negative powers3−236=3−2−6=3−8, 474− 3=47−(−3)=47+3=410
11th degree property When raising a power to a power, the powers are multiplied. ( an)m=an⋅m For example: (23)2=23⋅2=26=64
Table of powers up to 10
Few people manage to remember the entire table of degrees, and who needs it when it is so easy to find? Our power table includes both the popular tables of squares and cubes (from 1 to 10), as well as tables of other powers that are less common. The columns of the table of powers indicate the bases of the degree (the number that needs to be raised to a power), the rows indicate the exponents (the power to which the number needs to be raised), and at the intersection of the desired column and the desired row is the result of raising the desired number to a given power. There are several types of problems that can be solved using power tables. The immediate task is to calculate n th power of a number. The inverse problem, which can also be solved using a table of powers, may sound like this: “to what power should the number be raised? a to get the number b ?" or "What number to the power n gives a number b ?".
Table of powers up to 10
|
1 n |
2 n |
3 n |
4 n |
5 n |
6 n |
7 n |
8 n |
9 n |
10 n |
|
How to use the degree table
Let's look at a few examples of using the power table.
Example 1. What number results from raising the number 6 to the 8th power? In the table of degrees we look for column 6 n, since according to the conditions of the problem the number 6 is raised to a power. Then in the table of powers we look for line 8, since the given number must be raised to the power of 8. At the intersection we look at the answer: 1679616.
Example 2. To what power must the number 9 be raised to get 729? In the table of degrees we look for column 9 n and we go down it to the number 729 (the third line of our table of degrees). The line number is the required degree, that is, the answer: 3.
Example 3. What number must be raised to the power of 7 to get 2187? In the table of degrees we look for line 7, then move along it to the right to the number 2187. From the found number we go up and find out that the heading of this column is 3 n, which means the answer is: 3.
Example 4. To what power must the number 2 be raised to get 63? In the table of degrees we find column 2 n and we go down it until we meet 63... But this will not happen. We will never see the number 63 in this column or in any other column of the table of powers, which means that no integer from 1 to 10 gives the number 63 when raised to an integer power from 1 to 10. Thus, there is no answer .
Each arithmetic operation sometimes becomes too cumbersome to write and they try to simplify it. This was once the case with the addition operation. People needed to carry out repeated addition of the same type, 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 = 300. Due to its cumbersome nature, it was decided to shorten the notation to 3 * 100 = 300. In fact, the notation “three times one hundred” means that you need to take one hundred threes and add them together. Multiplication caught on and gained general popularity. But the world does not stand still, and in the Middle Ages the need arose to carry out repeated multiplication of the same type. I remember an old Indian riddle about a sage who asked for wheat grains in the following quantities as a reward for work done: for the first square of the chessboard he asked for one grain, for the second - two, for the third - four, for the fifth - eight, and so on. This is how the first multiplication of powers appeared, because the number of grains was equal to two to the power of the cell number. For example, on the last cell there would be 2*2*2*...*2 = 2^63 grains, which is equal to a number 18 characters long, which, in fact, is the meaning of the riddle.
The operation of exponentiation caught on quite quickly, and the need to carry out addition, subtraction, division and multiplication of powers also quickly arose. The latter is worth considering in more detail. The formulas for adding powers are simple and easy to remember. In addition, it is very easy to understand where they come from if the power operation is replaced by multiplication. But first you need to understand some basic terminology. The expression a^b (read “a to the power of b”) means that the number a should be multiplied by itself b times, with “a” being called the base of the power, and “b” the power exponent. If the bases of the degrees are the same, then the formulas are derived quite simply. Specific example: find the value of the expression 2^3 * 2^4. To know what should happen, you should find out the answer on the computer before starting the solution. Entering this expression into any online calculator, search engine, typing “multiplying powers with different bases and the same” or a mathematical package, the output will be 128. Now let’s write out this expression: 2^3 = 2*2*2, and 2^4 = 2 *2*2*2. It turns out that 2^3 * 2^4 = 2*2*2*2*2*2*2 = 2^7 = 2^(3+4) . It turns out that the product of powers with the same base is equal to the base raised to a power equal to the sum of the two previous powers.
You might think that this is an accident, but no: any other example can only confirm this rule. Thus, in general, the formula looks like this: a^n * a^m = a^(n+m) . There is also a rule that any number to the zero power is equal to one. Here we should remember the rule of negative powers: a^(-n) = 1 / a^n. That is, if 2^3 = 8, then 2^(-3) = 1/8. Using this rule, you can prove the validity of the equality a^0 = 1: a^0 = a^(n-n) = a^n * a^(-n) = a^(n) * 1/a^(n) , a^ (n) can be reduced and one remains. From here the rule is derived that the quotient of powers with the same bases is equal to this base to a degree equal to the quotient of the dividend and divisor: a^n: a^m = a^(n-m) . Example: simplify the expression 2^3 * 2^5 * 2^(-7) *2^0: 2^(-2) . Multiplication is a commutative operation, therefore, you must first add the multiplication exponents: 2^3 * 2^5 * 2^(-7) *2^0 = 2^(3+5-7+0) = 2^1 =2. Next you need to deal with division by a negative power. It is necessary to subtract the exponent of the divisor from the exponent of the dividend: 2^1: 2^(-2) = 2^(1-(-2)) = 2^(1+2) = 2^3 = 8. It turns out that the operation of dividing by a negative the degree is identical to the operation of multiplication by a similar positive exponent. So the final answer is 8.
There are examples where non-canonical multiplication of powers takes place. Multiplying powers with different bases is often much more difficult, and sometimes even impossible. Some examples of different possible techniques should be given. Example: simplify the expression 3^7 * 9^(-2) * 81^3 * 243^(-2) * 729. Obviously, there is a multiplication of powers with different bases. But it should be noted that all bases are different powers of three. 9 = 3^2.1 = 3^4.3 = 3^5.9 = 3^6. Using the rule (a^n) ^m = 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 = 3^7 * 3^(-4) * 3^(12) * 3^(-10) * 3^6 = 3^(7-4+12 -10+6) = 3^(11) . Answer: 3^11. In cases where there are different bases, the rule a^n * b^n = (a*b) ^n works for equal indicators. For example, 3^3 * 7^3 = 21^3. Otherwise, when the bases and exponents are different, complete multiplication cannot be performed. Sometimes you can partially simplify or resort to the help of computer technology.
Addition and subtraction of powers
It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.
So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.
Odds equal powers of identical variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is equal to 5a 2.
It is also obvious that if you take two squares a, or three squares a, or five squares a.
But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.
So, the sum of a 2 and a 3 is the sum of a 2 + a 3.
It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.
The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.
Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.
Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 — 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6
Multiplying powers
Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.
Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.
Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y
The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.
By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.
So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .
Here 5 is the power of the multiplication result, which is equal to 2 + 3, the sum of the powers of the terms.
So, a n .a m = a m+n .
For a n , a is taken as a factor as many times as the power of n;
And a m is taken as a factor as many times as the degree m is equal to;
That's why, powers with the same bases can be multiplied by adding the exponents of the powers.
So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .
Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (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 also true for numbers whose exponents are negative.
1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.
2. y -n .y -m = y -n-m .
3. a -n .a m = a m-n .
If a + b are multiplied by a - b, the result will be a 2 - b 2: that is
The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.
If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.
So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.
Division of degrees
Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.
Thus, a 3 b 2 divided by b 2 is equal to a 3.
Writing a 5 divided by a 3 looks like $\frac $. But this is equal to a 2 . In a series 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 by another, and the exponent will be equal to difference indicators of divisible numbers.
When dividing degrees with the same base, their exponents are subtracted..
So, y 3:y 2 = y 3-2 = y 1. That is, $\frac = y$.
And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.
Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3
The rule is also true for numbers with negative values of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.
h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$
It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.
Examples of solving examples with fractions containing numbers with powers
1. Decrease the exponents by $\frac $ Answer: $\frac $.
2. Decrease exponents by $\frac$. Answer: $\frac$ or 2x.
3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .
4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a 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 by (a - b)/3.
6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).
7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .
8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.
Properties of degree
We remind you that in this lesson we will understand properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.
A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.
Property No. 1
Product of powers
When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.
a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.
This property of powers also applies to the product of three or more powers.
- Simplify the expression.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15 - Present it as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17 - Present it as a degree.
(0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15 - Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2 - Calculate.
Please note that in the specified property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.
You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243
Property No. 2
Partial degrees
When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.
11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4
Answer: t = 3 4 = 81
Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.
Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5
Example. Find the value of an expression using the properties of exponents.
2 11 − 5 = 2 6 = 64
Please note that in Property 2 we were only talking about dividing powers with the same bases.
You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4
Property No. 3
Raising a degree to a power
When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.
(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.
We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.
How to multiply powers
How to multiply powers? Which powers can be multiplied and which cannot? How to multiply a number by a power?
In algebra, you can find a product of powers in two cases:
1) if the degrees have the same bases;
2) if the degrees have the same indicators.
When multiplying powers with the same bases, the base must be left the same, and the exponents must be added:
When multiplying degrees with the same indicators, the overall indicator can be taken out of brackets:
Let's look at how to multiply powers using specific examples.
The unit is not written in the exponent, but when multiplying powers, they take into account:
When multiplying, there can be any number of powers. It should be remembered that you don’t have to write the multiplication sign before the letter:
In expressions, exponentiation is done first.
If you need to multiply a number by a power, you should first perform the exponentiation, and only then the multiplication:
Multiplying powers with the same bases
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In this lesson we will study multiplication of powers with like bases. First, let us recall the definition of degree and formulate a theorem on the validity of the equality . Then we will give examples of its application on specific numbers and prove it. We will also apply the theorem to solve various problems.
Topic: Power with a natural exponent and its properties
Lesson: Multiplying powers with the same bases (formula)
1. Basic definitions
Basic definitions:
n- exponent,
— n th power of a number.
2. Statement of Theorem 1
Theorem 1. For any number A and any natural n And k the equality is true:
In other words: if A– any number; n And k natural numbers, then:
Hence rule 1:
3. Explanatory tasks
Conclusion: special cases confirmed the correctness of Theorem No. 1. Let us prove it in the general case, that is, for any A and any natural n And k.
4. Proof of Theorem 1
Given a number A– any; numbers n And k – natural. Prove:
The proof is based on the definition of degree.
5. Solving examples using Theorem 1
Example 1: Think of it as a degree.
To solve the following examples, we will use Theorem 1.
and) 
6. Generalization of Theorem 1
A generalization used here:
7. Solving examples using a generalization of Theorem 1
8. Solving various problems using Theorem 1
Example 2: Calculate (you can use the table of basic powers).
A) 
(according to the table)
b) 
Example 3: Write it as a power with base 2.
A) 
Example 4: Determine the sign of the number:
, A - negative, since the exponent at -13 is odd.
Example 5: Replace (·) with a power of a number with a base r:
We have, that is.
9. Summing up
1. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 7. 6th edition. M.: Enlightenment. 2010
1. School assistant (Source).
1. Present as a power:
a B C D E) 
3. Write as a power with base 2:
4. Determine the sign of the number:
A) 
5. Replace (·) with a power of a number with a base r:
a) r 4 · (·) = r 15; b) (·) · r 5 = r 6
Multiplication and division of powers with the same exponents
In this lesson we will study multiplication of powers with equal exponents. First, let's recall the basic definitions and theorems about multiplying and dividing powers with the same bases and raising powers to powers. Then we formulate and prove theorems on multiplication and division of powers with the same exponents. And then with their help we will solve a number of typical problems.
Reminder of basic definitions and theorems
Here a- the basis of the degree,
— n th power of a number.
Theorem 1. For any number A and any natural n And k the equality is true:
When multiplying powers with the same bases, the exponents are added, the base remains unchanged.
Theorem 2. For any number A and any natural n And k, such that n > k the equality is true:
When dividing degrees with the same bases, the exponents are subtracted, but the base remains unchanged.
Theorem 3. For any number A and any natural n And k the equality is true:
All the theorems listed were about powers with the same reasons, in this lesson we will look at degrees with the same indicators.
Examples for multiplying powers with the same exponents
Consider the following examples:
Let's write down the expressions for determining the degree.
Conclusion: From the examples it can be seen that 
, but this still needs to be proven. Let us formulate the theorem and prove it in the general case, that is, for any A And b and any natural n.
Formulation and proof of Theorem 4
For any numbers A And b and any natural n the equality is true:
Proof Theorem 4 .
By definition of degree:
So we have proven that .
To multiply powers with the same exponents, it is enough to multiply the bases and leave the exponent unchanged.
Formulation and proof of Theorem 5
Let us formulate a theorem for dividing powers with the same exponents.
For any number A And b() and any natural n the equality is true:
Proof Theorem 5 .
Let's write down the definition of degree:
Statement of theorems in words
So, we have proven that .
To divide powers with the same exponents into each other, it is enough to divide one base by another, and leave the exponent unchanged.
Solving typical problems using Theorem 4
Example 1: Present as a product of powers.
To solve the following examples, we will use Theorem 4.
To solve the following example, recall the formulas:
Generalization of Theorem 4
Generalization of Theorem 4:
Solving Examples Using Generalized Theorem 4
Continuing to solve typical problems
Example 2: Write it as a power of the product.
Example 3: Write it as a power with exponent 2.
Calculation examples
Example 4: Calculate in the most rational way.
2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: VENTANA-GRAF
3. Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E. and others. Algebra 7.M.: Enlightenment. 2006
2. School assistant (Source).
1. Present as a product of powers:
A) ; b) ; V) ; G) ;
2. Write as a power of the product:
3. Write as a power with exponent 2:
4. Calculate in the most rational way.
Mathematics lesson on the topic “Multiplication and division of powers”
Sections: Mathematics
Pedagogical goal:
Tasks:
Activity units of teaching: determination of degree with a natural indicator; degree components; definition of private; combinational law of multiplication.
I. Organizing a demonstration of students’ mastery of existing knowledge. (step 1)
a) Updating knowledge:
2) Formulate a definition of degree with a natural exponent.
a n =a a a a … a (n times)
b k =b b b b a… b (k times) Justify the answer.
II. Organization of self-assessment of the student’s degree of proficiency in current experience. (step 2)
Self-test: (individual work in two versions.)
A1) Present the product 7 7 7 7 x x x as a power:
A2) Represent the power (-3) 3 x 2 as a product
A3) Calculate: -2 3 2 + 4 5 3
I select the number of tasks in the test in accordance with the preparation of the class level.
I give you the key to the test for self-test. Criteria: pass - no pass.
III. Educational and practical task (step 3) + step 4. (the students themselves will formulate the properties)
While solving problems 1) and 2), students propose a solution, and I, as a teacher, organize the class to find a way to simplify powers when multiplying with the same bases.
Teacher: come up with a way to simplify powers when multiplying with the same bases.
An entry appears on the cluster:
The topic of the lesson is formulated. Multiplication of powers.
Teacher: come up with a rule for dividing powers with the same bases.
Reasoning: what action is used to check division? a 5: a 3 = ? that a 2 a 3 = a 5
I return to the diagram - a cluster and add to the entry - .. when dividing, we subtract and add the topic of the lesson. ...and division of degrees.
IV. Communicating to students the limits of knowledge (as a minimum and as a maximum).
Teacher: the minimum task for today’s lesson is to learn to apply the properties of multiplication and division of powers with the same bases, and the maximum task is to apply multiplication and division together.
We write on the board : a m a n = a m+n ; a m: a n = a m-n
V. Organization of studying new material. (step 5)
a) According to the textbook: No. 403 (a, c, e) tasks with different wordings
No. 404 (a, d, f) independent work, then I organize a mutual check, give the keys.
b) For what value of m is the equality valid? a 16 a m = a 32; x h x 14 = x 28; x 8 (*) = x 14
Assignment: come up with similar examples for division.
c) No. 417 (a), No. 418 (a) Traps for students: x 3 x n = x 3n; 3 4 3 2 = 9 6 ; a 16: a 8 = a 2.
VI. Summarizing what has been learned, conducting diagnostic work (which encourages students, and not the teacher, to study this topic) (step 6)
Diagnostic work.
Test(place the keys on the back of the dough).
Task options: represent the quotient x 15 as a power: x 3; represent as a power the product (-4) 2 (-4) 5 (-4) 7 ; for which m is the equality a 16 a m = a 32 valid? find the value of the expression h 0: h 2 at h = 0.2; calculate the value of the expression (5 2 5 0) : 5 2 .
Lesson summary. Reflection. I divide the class into two groups.
Find arguments in group I: in favor of knowing the properties of the degree, and group II - arguments that will say that you can do without properties. We listen to all the answers and draw conclusions. In subsequent lessons, you can offer statistical data and call the rubric “It’s beyond belief!”
VII. Homework.
Historical reference. What numbers are called Fermat numbers.
P.19. No. 403, No. 408, No. 417
Used Books:
Properties of degrees, formulations, proofs, examples.
After the power of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.
Page navigation.
Properties of degrees with natural exponents
By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, we can obtain and justify the following properties of degree with natural exponent:
- if a>0, then a n>0 for any natural number n;
- if a=0, then a n =0;
- if a 2·m >0 , if a 2·m−1 n ;
- if m and n are natural numbers such that m>n, then for 0m n, and for a>0 the inequality a m >a n is true.
- a m ·a n =a m+n ;
- a m:a n =a m−n ;
- (a·b) n =a n ·b n ;
- (a:b) n =a n:b n ;
- (a m) n =a m·n ;
- if n is a positive integer, a and b are positive numbers, and a n n and a −n >b −n ;
- if m and n are integers, and m>n, then for 0m n, and for a>1 the inequality a m >a n holds.
Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying expressions often used in the form a m+n =a m ·a n .
Now let's look at each of them in detail.
Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.
Let us prove the main property of the degree. By the definition of a power with a natural exponent, the product of powers with identical bases of the form a m ·a n can be written as the product 
. Due to the properties of multiplication, the resulting expression can be written as 
, and this product is a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.
Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values of the expressions 2 2 · 2 3 and 2 5 . Carrying out exponentiation, we have 2 2 2 3 =(2 2) (2 2 2) = 4 8 = 32 and 2 5 =2 2 2 2 2 = 32 , since we get equal values, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.
The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1 , n 2 , …, n k the equality a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k is true.
For example, (2,1) 3 ·(2,1) 3 ·(2,1) 4 ·(2,1) 7 = (2,1) 3+3+4+7 =(2,1) 17.
We can move on to the next property of powers with a natural exponent – property of quotient powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n, the equality a m:a n =a m−n is true.
Before presenting the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that we cannot divide by zero. The condition m>n is introduced so that we do not go beyond the natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n) or a negative number (which happens for m m−n ·a n =a (m−n) +n =a m. From the resulting equality a m−n ·a n =a m and from the connection between multiplication and division it follows that a m−n is a quotient of powers a m and an n. This proves the property of quotients of powers with the same bases.
Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the equality π 5:π 2 =π 5−3 =π 3 corresponds to the considered property of the degree.
Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n .
Indeed, by the definition of a degree with a natural exponent we have 
. Based on the properties of multiplication, the last product can be rewritten as 
, which is equal to a n · b n .
Here's an example: 
.
This property extends to the power of the product of three or more factors. That is, the property of natural degree n of a product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n .
For clarity, we will show this property with an example. For the product of three factors to the power of 7 we have .
The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of powers a n and b n, that is, (a:b) n =a n:b n.
The proof can be carried out using the previous property. So (a:b) n ·b n =((a:b)·b) n =a n , and from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of division a n on bn.
Let's write this property using specific numbers as an example: 
.
Now let's voice it property of raising a power to a power: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of the number a with exponent m·n, that is, (a m) n =a m·n.
For example, (5 2) 3 =5 2·3 =5 6.
The proof of the power-to-degree property is the following chain of equalities: 
.
The property considered can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality 
. For greater clarity, let's give an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10.
It remains to dwell on the properties of comparing degrees with a natural exponent.
Let's start by proving the property of comparing zero and power with a natural exponent.
First, let's prove that a n >0 for any a>0.
The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a, the degree a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and 
.
It is quite obvious that for any natural number n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0.
Let's move on to negative bases of degree.
Let's start with the case when the exponent is an even number, let's denote it as 2·m, where m is a natural number. Then 
. According to the rule for multiplying negative numbers, each of the products of the form a·a is equal to the product of the absolute values of the numbers a and a, which means that it is a positive number. Therefore, the product will also be positive 
and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .
Finally, when the base a is a negative number and the exponent is an odd number 2 m−1, then 
. All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3 17 n n is the product of the left and right sides of n true inequalities a properties of inequalities, a provable inequality of the form a n n is also true. For example, due to this property, the inequalities 3 7 7 and 
.
It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of two powers with natural exponents and identical positive bases less than one, 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 proceed to the proof of this property.
Let us prove that for m>n and 0m n . To do this, we write down the difference a m − a n and compare it with zero. The recorded difference, after taking a n out of brackets, will take the form a n ·(a m−n−1) . The resulting product is negative as the product of a positive number a n and a negative number a m−n −1 (a n is positive as the natural power of a positive number, and the difference a m−n −1 is negative, since m−n>0 due to the initial condition m>n, whence it follows that when 0m−n is less than unity). Therefore, a m −a n m n , which is what needed to be proven. As an example, we give the correct inequality.
It remains to prove the second part of the property. Let us prove that for m>n and a>1 a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1 the degree a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.
Properties of powers with integer exponents
Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.
We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.
So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true: properties of powers with integer exponents:
When a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.
Proving each of these properties is not difficult; to do this, it is enough to use the definitions of degrees with natural and integer exponents, as well as the properties of operations with real numbers. As an example, let us prove that the power-to-power property holds for both positive integers and non-positive integers. To do this, you need to show that if p is zero or a natural number and q is zero or a natural number, then 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) . Let's do it.
For positive p and q, the equality (a p) q =a p·q was proven in the previous paragraph. If p=0, then we have (a 0) q =1 q =1 and a 0·q =a 0 =1, whence (a 0) q =a 0·q. Similarly, if q=0, then (a p) 0 =1 and a p·0 =a 0 =1, whence (a p) 0 =a p·0. If both p=0 and q=0, then (a 0) 0 =1 0 =1 and a 0·0 =a 0 =1, whence (a 0) 0 =a 0·0.
Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then 
. By the property of quotients to powers we have 
. Since 1 p =1·1·…·1=1 and , then . The last expression, by definition, is a power of the form a −(p·q), which, due to the rules of multiplication, can be written as a (−p)·q.
Likewise 
.
AND 
.
Using the same principle, you can prove all other properties of a degree with an integer exponent, written in the form of equalities.
In the penultimate of the recorded properties, it is worth dwelling on the proof of the inequality a −n >b −n, which is valid for any negative integer −n and any positive a and bfor which the condition a is satisfied . Let us write down and transform the difference between the left and right sides of this inequality: 
. Since by condition a n n , therefore, b n −a n >0 . The product a n · b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as the quotient of the positive numbers b n −a n and a n ·b n . Therefore, whence a −n >b −n , which is what needed to be proved.
The last property of powers with integer exponents is proved in the same way as a similar property of powers with natural exponents.
Properties of powers with rational exponents
We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers with integer exponents. Namely:
- property of the product of powers with the same bases
for a>0, and if and, then for a≥0; - property of quotient powers with the same bases
for a>0 ; - property of a product to a fractional power
for a>0 and b>0, and if and, then for a≥0 and (or) b≥0; - property of a quotient to a fractional power
for a>0 and b>0, and if , then for a≥0 and b>0; - property of degree to degree
for a>0, and if and, then for a≥0; - property of comparing powers with equal rational exponents: for any positive numbers a and b, a 0 the inequality a p p is true, and for p p >b p ;
- the property of comparing powers with rational exponents and equal bases: for rational numbers p and q, p>q for 0p q, and for a>0 – inequality a p >a q.
- a p ·a q =a p+q ;
- a p:a q =a p−q ;
- (a·b) p =a p ·b p ;
- (a:b) p =a p:b p ;
- (a p) q =a p·q ;
- for any positive numbers a and b, a 0 the inequality a p p is true, and for p p >b p ;
- for irrational numbers p and q, p>q for 0p q, and for a>0 – the inequality a p >a q.
The proof of the properties of powers with fractional exponents is based on the definition of a power with a fractional exponent, on the properties of the arithmetic root of the nth degree and on the properties of a power with an integer exponent. Let us provide evidence.
By definition of a power with a fractional exponent and , then 
. The properties of the arithmetic root allow us to write the following equalities. Further, using the property of a degree with an integer exponent, we obtain , from which, by the definition of a degree with a fractional exponent, we have 
, and the indicator of the degree obtained can be transformed as follows: . This completes the proof.
The second property of powers with fractional exponents is proved in an absolutely similar way: 
The remaining equalities are proved using similar principles: 
Let's move on to proving the next property. Let us prove that for any positive a and b, a 0 the inequality a p p is true, and for p p >b p . Let's write the rational number p as m/n, where m is an integer and n is a natural number. The conditions p 0 in this case will be equivalent to the conditions m 0, respectively. For m>0 and am m . From this inequality, by the property of roots, we have, and since a and b are positive numbers, then, based on the definition of a degree with a fractional exponent, the resulting inequality can be rewritten as, that is, a p p .
Similarly, for m m >b m , whence, that is, a p >b p .
It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q, p>q for 0p q, and for a>0 – the inequality a p >a q. We can always reduce rational numbers p and q to a common denominator, even if we get ordinary fractions and , where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from the rule for comparing ordinary fractions with the same denominators. Then, by the property of comparing degrees with the same bases and natural exponents, for 0m 1 m 2, and for a>1, the inequality a m 1 >a m 2. These inequalities in the properties of the roots can be rewritten accordingly as 
And 
. And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From here we draw the final conclusion: for p>q and 0p q , and for a>0 – the inequality a p >a q .
Properties of powers with irrational exponents
From the way a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0, b>0 and irrational numbers p and q the following are true properties of powers with irrational exponents:
From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.
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After the power of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.
Page navigation.
Properties of degrees with natural exponents
By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, we can obtain and justify the following properties of degree with natural exponent:
- the main property of the degree a m ·a n =a m+n, its generalization;
- property of quotient powers with identical bases a m:a n =a m−n ;
- product power property (a·b) n =a n ·b n , its extension;
- property of the quotient to the natural degree (a:b) n =a n:b n ;
- raising a degree to a power (a m) n =a m·n, its generalization (((a n 1) n 2) …) n k =a n 1 ·n 2 ·…·n k;
- comparison of degree with zero:
- if a>0, then a n>0 for any natural number n;
- if a=0, then a n =0;
- if a<0 и показатель степени является четным числом 2·m , то a 2·m >0 if a<0 и показатель степени есть нечетное число 2·m−1 , то a 2·m−1 <0 ;
- if a and b are positive numbers and a
- if m and n are natural numbers such that m>n , then at 0
- if m and n are natural numbers such that m>n , then at 0
Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying expressions often used in the form a m+n =a m ·a n .
Now let's look at each of them in detail.
Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.
Let us prove the main property of the degree. By the definition of a power with a natural exponent, the product of powers with the same bases of the form a m ·a n can be written as a product. Due to the properties of multiplication, the resulting expression can be written as 
, and this product is a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.
Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values of the expressions 2 2 · 2 3 and 2 5 . Performing exponentiation, we have 2 2 ·2 3 =(2·2)·(2·2·2)=4·8=32 and 2 5 =2·2·2·2·2=32, since equal values are obtained, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.
The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1, n 2, …, n k the following equality is true: a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k.
For example, (2,1) 3 ·(2,1) 3 ·(2,1) 4 ·(2,1) 7 = (2,1) 3+3+4+7 =(2,1) 17 .
We can move on to the next property of powers with a natural exponent – property of quotient powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n, the equality a m:a n =a m−n is true.
Before presenting the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that we cannot divide by zero. The condition m>n is introduced so that we do not go beyond the natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n ) or a negative number (which happens for m Proof. The main property of a fraction allows us to write the equality a m−n ·a n =a (m−n)+n =a m. From the resulting equality a m−n ·a n =a m and it follows that a m−n is a quotient of the powers a m and a n . This proves the property of quotient powers with identical bases. Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the equality π 5:π 2 =π 5−3 =π 3 corresponds to the considered property of the degree. Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n . Indeed, by the definition of a degree with a natural exponent we have Here's an example: This property extends to the power of the product of three or more factors. That is, the property of natural degree n of the product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n. For clarity, we will show this property with an example. For the product of three factors to the power of 7 we have . The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of powers a n and b n, that is, (a:b) n =a n:b n. The proof can be carried out using the previous property. So (a:b) n b n =((a:b) b) n =a n, and from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of a n divided by b n . Let's write this property using specific numbers as an example: Now let's voice it property of raising a power to a power: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of the number a with exponent m·n, that is, (a m) n =a m·n. For example, (5 2) 3 =5 2·3 =5 6. The proof of the power-to-degree property is the following chain of equalities: The property considered can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality It remains to dwell on the properties of comparing degrees with a natural exponent. Let's start by proving the property of comparing zero and power with a natural exponent. First, let's prove that a n >0 for any a>0. The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a, the degree a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and It is quite obvious that for any natural number n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0. Let's move on to negative bases of degree. Let's start with the case when the exponent is an even number, let's denote it as 2·m, where m is a natural number. Then Finally, when the base a is a negative number and the exponent is an odd number 2 m−1, then Let's move on to the property of comparing powers with the same natural exponents, which has the following formulation: of two powers with the same natural exponents, n is less than the one whose base is smaller, and greater is the one whose base is greater. Let's prove it. Inequality a n properties of inequalities a provable inequality of the form a n is also true (2.2) 7 and It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of two powers with natural exponents and identical positive bases less than one, 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 proceed to the proof of this property. Let us prove that for m>n and 0 It remains to prove the second part of the property. Let us prove that for m>n and a>1 a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1 the degree a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.
. Based on the properties of multiplication, the last product can be rewritten as , which is equal to a n · b n .
.
..
. For greater clarity, here is an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10
.
.
. For each of the products of the form a·a is equal to the product of the moduli of the numbers a and a, which means it is a positive number. Therefore, the product will also be positive 
and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .. All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3<0
, (−0,003) 17 <0
и 
.
.
Properties of powers with integer exponents
Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.
We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.
So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true: properties of powers with integer exponents:
- a m ·a n =a m+n ;
- a m:a n =a m−n ;
- (a·b) n =a n ·b n ;
- (a:b) n =a n:b n ;
- (a m) n =a m·n ;
- if n is a positive integer, a and b are positive numbers, and a b−n ;
- if m and n are integers, and m>n , then at 0
When a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.
Proving each of these properties is not difficult; to do this, it is enough to use the definitions of degrees with natural and integer exponents, as well as the properties of operations with real numbers. As an example, let us prove that the power-to-power property holds for both positive integers and non-positive integers. To do this, you need to show that if p is zero or a natural number and q is zero or a natural number, then 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). Let's do it.
For positive p and q, the equality (a p) q =a p·q was proven in the previous paragraph. If p=0, then we have (a 0) q =1 q =1 and a 0·q =a 0 =1, whence (a 0) q =a 0·q. Similarly, if q=0, then (a p) 0 =1 and a p·0 =a 0 =1, whence (a p) 0 =a p·0. If both p=0 and q=0, then (a 0) 0 =1 0 =1 and a 0·0 =a 0 =1, whence (a 0) 0 =a 0·0.
Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then . By the property of quotients to powers we have 
. Since 1 p =1·1·…·1=1 and , then . The last expression, by definition, is a power of the form a −(p·q), which, due to the rules of multiplication, can be written as a (−p)·q.
Likewise 
.
AND 
.
Using the same principle, you can prove all other properties of a degree with an integer exponent, written in the form of equalities.
In the penultimate of the recorded properties, it is worth dwelling on the proof of the inequality a −n >b −n, which is valid for any negative integer −n and any positive a and bfor which the condition a is satisfied . Since by condition a 0 . The product a n · b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as the quotient of the positive numbers b n −a n and a n ·b n . Therefore, whence a −n >b −n , which is what needed to be proved.
The last property of powers with integer exponents is proved in the same way as a similar property of powers with natural exponents.
Properties of powers with rational exponents
We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers with integer exponents. Namely:
The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, and on the properties of a degree with an integer exponent. Let us provide evidence.
By definition of a power with a fractional exponent and , then 
. The properties of the arithmetic root allow us to write the following equalities. Further, using the property of a degree with an integer exponent, we obtain , from which, by the definition of a degree with a fractional exponent, we have , and the indicator of the degree obtained can be transformed as follows: . This completes the proof.
The second property of powers with fractional exponents is proved in an absolutely similar way: 
The remaining equalities are proved using similar principles: 
Let's move on to proving the next property. Let us prove that for any positive a and b, a b p . Let's write the rational number p as m/n, where m is an integer and n is a natural number. Conditions p<0 и p>0 in this case the conditions m<0 и m>0 accordingly. For m>0 and a
Similarly, for m<0 имеем a m >b m , from where, that is, and a p >b p .
It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q, p>q at 0 And 
. And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From here we draw the final conclusion: for p>q and 0
Properties of powers with irrational exponents
From the way a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0, b>0 and irrational numbers p and q the following are true properties of powers with irrational exponents:
- a p ·a q =a p+q ;
- a p:a q =a p−q ;
- (a·b) p =a p ·b p ;
- (a:b) p =a p:b p ;
- (a p) q =a p·q ;
- for any positive numbers a and b, a 0 the inequality a p b p ;
- for irrational numbers p and q, p>q at 0
From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.
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