Basic Mathematics for College Students

Multiply exponential expressions that have like bases.

To develop a rule for multiplying exponential expressions that have the same base, we

consider the product. Since means that 6 is to be used as a factor two times,

and means that 6 is to be used as a factor three times, we have

2 factors of 6 3 factors of 6

5 factors of 6

We can quickly find this result if we keep the common base 6 and add the exponents

on and.

This example illustrates the following rule for exponents.

Product Rule for Exponents

To multiply exponential expressions that have the same base, keep the

common base and add the exponents.

For any number and any natural numbers and ,

Read as “ to the th power times to the

nth power equals to the x mplus th power.”n

xmxnxmn x m x

x m n

62  63  62 ^3  65

62 63

 65

 6  6  6  6  6

⎫⎪⎪⎬⎪⎪⎭

62  63 ^6  6  6  6  6

⎫⎬⎭ ⎫⎬⎭

63

62  63 62

2

690 Chapter 8 An Introduction to Algebra

Self Check 3

Simplify:

a.

b.

c.

d.

Now TryProblems 33, 35, and 37

(s^4 t^3 )(s^4 t^4 )

(y1)^5 (y1)^5

x^2 x^3 x

78 (7^7 )

EXAMPLE (^3) Simplify:
a. b. c. d.
StrategyIn each case, we want to write an equivalent expression using one base
and one exponent. We will use the product rule for exponents to do this.
WHYThe product rule for exponents is used to multiply exponential expressions
that have the same base.
Solution
a. Keep the common base, 9, and add the exponents.
Since is a very large number, we will leave the
answer in this form. We won’t evaluate it.
Caution! Don’t make the mistake of multiplying the bases when using the
product rule. Keep the samebase.
b. Keep the common base, x, and add the exponents.
c. Write yas y^1.
Keep the common base, y, and add the exponents.
d. Use the commutative and associative properties of
multiplication to group like bases together.
Keep the common base, c, and add the exponents.
Keep the common base, d, and add the exponents.
c^6 d^8
(c^2 ^4 )(d^3 ^5 )
(c^2 d^3 )(c^4 d^5 )(c^2 c^4 )(d^3 d^5 )
y^7
y^2 ^4 ^1
y^2 y^4 yy^2 y^4 y^1
x^3 x^4 x^3 ^4 x^7
95 (9^6 ) 8111
911

95 (9^6 ) 95 ^6  911

95 (9^6 ) x^3 x^4 y^2 y^4 y (c^2 d^3 )(c^4 d^5 )