Basic Mathematics for College Students

Find powers of products.

To develop another rule for exponents, we consider the expression , which is a

power of the productof 2 and.

Write the base 2x as a factor 3 times.

Change the order of the factors and group like bases.

Write each product of repeated factors in exponential form.

Evaluate:.

This example illustrates the following rule for exponents.

Power of a Product

To raise a product to a power, raise each factor of the product to that power.

For any numbers and , and any natural number ,

(xy)nxnyn

x y n

 8 x^323  8

 23 x^3

(2 2 2)(xxx)

(2x)^3  2 x 2 x 2 x

x

(2x)^3

4

692 Chapter 8 An Introduction to Algebra

Solution

a. Keep the base, 2, and multiply the exponents. Since 2^21 is a very

large number, we will leave the answer in this form.

b. Keep the base, , and multiply the exponents.

Since is a very large number, we will leave

the answer in this form.

c.(z^8 )^8 z^8 ^8 z^64 Keep the base, , and multiply the exponents.z

(6)^10

[(6)^2 ]^5 (6)^2 ^5 (6)^10  6

(2^3 )^7  23 ^7  221

EXAMPLE (^5) Simplify: a. b.
StrategyIn each case, we want to write an equivalent expression using one base
and one exponent. We will use the product and power rules for exponents to do this.
WHYThe expressions involve multiplication of exponential expressions that have
the same base and they involve powers of powers.
Solution
a. Within the parentheses, keep the common base, x, and add the
exponents: 2  5 7.
Keep the base, x, and multiply the exponents: 7  2 14.
b. For each power of zraised to a power, keep the base
and multiply the exponents: 2  4 8 and 3  3 9.
z^17 Keep the common base, z, and add the exponents: 8  9 17.
(z^2 )^4 (z^3 )^3 z^8 z^9
x^14
(x^2 x^5 )^2 (x^7 )^2
(x^2 x^5 )^2 (z^2 )^4 (z^3 )^3
Self Check 5
Simplify:
a.
b.
Now TryProblems 57 and 61
(a^3 )^3 (a^4 )^2
(a^4 a^3 )^3
EXAMPLE (^6) Simplify: a. b.
StrategyIn each case, we want to write the expression in an equivalent form in
which each base is raised to a single power. We will use the power of a product rule
for exponents to do this.
WHYWithin each set of parentheses is a product, and each of those products is
raised to a power.
(3c)^4 (x^2 y^3 )^5
Self Check 6
Simplify:
a.
b.
Now TryProblems 65 and 69
(c^3 d^4 )^6
(2t)^4