Basic Mathematics for College Students

Caution! We cannot use the product rule to simplify expressions like ,

where the bases are not the same. However, we can simplify this expression by

doing the arithmetic:

32  3  3 9 and 2^3  2  2  2 8.

Recall that like termsare terms with exactly the same variables raised to exactly
the same powers. To add or subtract exponential expressions, they must be like terms.
To multiply exponential expressions, only the bases need to be the same.

These are not like terms; the exponents are different. We cannot add.

These are like terms; we can add. Recall that.

The bases are the same; we can multiply.

Raise exponential expressions to a power.

To develop another rule for exponents, we consider. Here, an exponential
expression, , is raised to a power. Since is the base and 4 is the exponent, can
be written as. Because each of the four factors of contains three
factors of 5, there are or 12 factors of 5.

12 factors of 5

We can quickly find this result if we keep the common base of 5 and multiply the
exponents.

This example illustrates the following rule for exponents.

Power Rule for Exponents

To raise an exponential expression to a power, keep the base and multiply the

exponents.

For any number and any natural numbers and ,

Read as “the quantity of to the th power raised

to the th power equals to the th power.”

The Language of Algebra An exponential expression raised to a power, such

as (5^3 )^4 , is also called a power of a power.

n x mn

(xm)nxm^ ^ nxmn x m

x m n

(5^3 )^4  53 ^4  512

53 53 53 53

(5 ⎫⎬⎭ ⎫⎬⎭ ⎫⎬⎭ ⎫⎬⎭

(^3) ) (^4)  53  53  53  53  5  5  5  5  5  5  5  5  5  5  5  5  512

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

4  3

53  53  53  53 53

53 53 (5^3 )^4

(5^3 )^4

3

x^5 x^2 x^7

x^2 x^2  2 x^2 x^2  1 x^2

x^5 x^2

32  23  9  8  72

32  23

8.6 Multiplication Rules for Exponents 691

EXAMPLE (^4) Simplify: a. b. c.
StrategyIn each case, we want to write an equivalent expression using one base
and one exponent. We will use the power rule for exponents to do this.
WHYEach expression is a power of a power.
(2^3 )^7 [(6)^2 ]^5 (z^8 )^8
Self Check 4
Simplify:
a.
b.
Now TryProblems 49, 51, and 53
(y^5 )^2

(4^6 )^5