Is it possible to multiply numbers with the same powers? How to multiply powers, multiplying powers with different exponents

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 cell 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 basis 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 view 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 negative degree. 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. There are a few examples of different possible techniques. 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 reasons are various degrees threes. 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 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 the bases are different, 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.

In the last video lesson, we learned that the degree of a certain base is an expression that represents the product of the base by itself, taken in an amount equal to the exponent. Let's now study some the most important properties and operations of degrees.

For example, let's multiply two different degrees with the same base:

Let's present this work in its entirety:

(2) 3 * (2) 2 = (2)*(2)*(2)*(2)*(2) = 32

Having calculated the value of this expression, we get the number 32. On the other hand, as can be seen from the same example, 32 can be represented as the product of the same base (two), taken 5 times. And indeed, if you count it, then:

Thus, we can confidently conclude that:

(2) 3 * (2) 2 = (2) 5

This rule works successfully for any indicators and any reasons. This property of power multiplication follows from the rule that the meaning of expressions is preserved during transformations in a product. For any base a, the product of two expressions (a)x and (a)y is equal to a(x + y). In other words, when any expressions with the same base are produced, the resulting monomial has a total degree formed by adding the degrees of the first and second expressions.

The presented rule also works great when multiplying several expressions. The main condition is that everyone has the same bases. For example:

(2) 1 * (2) 3 * (2) 4 = (2) 8

It is impossible to add degrees, and indeed to carry out any power-based joint actions with two elements of an expression if their bases are different.
As our video shows, due to the similarity of the processes of multiplication and division, the rules for adding powers in a product are perfectly transferred to the division procedure. Consider this example:

Let us carry out a term-by-term transformation of the expression into full view and reduce the same elements in the dividend and divisor:

(2)*(2)*(2)*(2)*(2)*(2) / (2)*(2)*(2)*(2) = (2)(2) = (2) 2 = 4

The end result of this example is not so interesting, because already in the process of solving it it is clear that the value of the expression is equal to the square of two. And it is two that is obtained by subtracting the degree of the second expression from the degree of the first.

To determine the degree of the quotient, it is necessary to subtract the degree of the divisor from the degree of the dividend. The rule works if the basis is the same for all its values ​​and for all natural degrees. In the form of abstraction we have:

(a) x / (a) y = (a) x - y

From the rule of dividing identical bases with degrees, the definition for the zero degree follows. Obviously, the following expression looks like:

(a) x / (a) x = (a) (x - x) = (a) 0

On the other hand, if we do the division in a more visual way, we get:

(a) 2 / (a) 2 = (a) (a) / (a) (a) = 1

When reducing all visible elements of a fraction, the expression 1/1 is always obtained, that is, one. Therefore, it is generally accepted that any base raised to the zero power is equal to one:

Regardless of the value of a.

However, it would be absurd if 0 (which still gives 0 for any multiplication) is somehow equal to one, so an expression of the form (0) 0 (zero to the zero power) simply does not make sense, and to formula (a) 0 = 1 add a condition: “if a is not equal to 0.”

Let's solve the exercise. Let's find the meaning of the expression:

(34) 7 * (34) 4 / (34) 11

Since the base is the same everywhere and equal to 34, the final value will have the same base with a degree (according to the above rules):

In other words:

(34) 7 * (34) 4 / (34) 11 = (34) 0 = 1

Answer: the expression is equal to one.

Degree formulas used in the process of reduction and simplification complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a When:

Operations with degrees.

1. By multiplying degrees with the same base, their indicators are added:

a m·a n = a m + n .

2. When dividing degrees with the same base, their exponents are subtracted:

3. Power of the product of 2 or more factors is equal to the product of the powers of these factors:

(abc…) n = a n · b n · c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n /b n .

5. Raising a power to a power, the exponents are multiplied:

(a m) n = a m n .

Each formula above is true in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the radical number to this power:

4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:

5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:

Degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:

Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.

A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.

If you need to raise a specific number to a power, you can use . Now we will take a closer look at properties of degrees.

Exponential numbers open up great possibilities, they allow us to transform multiplication into addition, and adding is much easier than multiplying.

For example, we need to multiply 16 by 64. The product of multiplying these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. That is, 16 by 64 = 4x4x4x4x4, which is also equal to 1024.

The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.

Now let's use the rule. 16=4 2, or 2 4, 64=4 3, or 2 6, at the same time 1024=6 4 =4 5, or 2 10.

Therefore, our problem can be written differently: 4 2 x4 3 =4 5 or 2 4 x2 6 =2 10, and each time we get 1024.

We can solve a number of similar examples and see that multiplying numbers with powers reduces to adding exponents, or exponential, of course, provided that the bases of the factors are equal.

Thus, without performing multiplication, we can immediately say that 2 4 x2 2 x2 14 = 2 20.

This rule is also valid when dividing numbers with powers, but in this case the exponent of the divisor is subtracted from the exponent of the dividend. Thus, 2 5:2 3 =2 2, which in ordinary numbers is equal to 32:8 = 4, that is, 2 2. Let's summarize:

a m x a n =a m+n, a m: a n =a m-n, where m and n are integers.

At first glance it may seem that this is multiplying and dividing numbers with powers not very convenient, because first you need to represent the number in exponential form. It is not difficult to represent the numbers 8 and 16, that is, 2 3 and 2 4, in this form, but how to do this with the numbers 7 and 17? Or what to do in cases where a number can be represented in exponential form, but the bases for exponential expressions of numbers are very different. For example, 8x9 is 2 3 x 3 2, in which case we cannot sum the exponents. Neither 2 5 nor 3 5 are the answer, nor does the answer lie in the interval between these two numbers.

Then is it worth bothering with this method at all? Definitely worth it. It provides enormous benefits, especially for complex and time-consuming calculations.

Lesson on the topic: "Rules of multiplication and division of powers with the same and different exponents. Examples"

Additional materials
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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 become familiar 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 took 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 powers are multiplied with different reasons, but with the same indicator.
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 $\frac(a^n)( b^n)$ is necessary. 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.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.