Introduction

To multiply powers of the same base, we simply add the exponents. This is called the product rule for exponents. For example, in the expression

{\color{blue}3}^4 \times {\color{blue}3}^2,

both powers have the same base ({\color{blue}3}). Therefore, we can simplify this expression by adding the exponents:

{\color{blue}3}^4 \times {\color{blue}3}^2 = {\color{blue}3}^{4+2} = {\color{blue}3}^6

Note: The product rule for exponents is really just a quicker alternative to using repeated multiplication, as follows:

\begin{align} {\color{blue}3}^4 \times {\color{blue}3}^2 &= \\ \underbrace{ {\color{blue}3} \times {\color{blue}3} \times {\color{blue}3} \times {\color{blue}3} }_{ 4 \text{ copies} } \times \underbrace{ {\color{blue}3} \times {\color{blue}3} }_{ 2 \text{ copies} } &= \\ \underbrace{ {\color{blue}3} \times {\color{blue}3} \times {\color{blue}3} \times {\color{blue}3} \times {\color{blue}3} \times {\color{blue}3} }_{ 6 \text{ copies} } &= \\ {\color{blue}3}^6 \end{align}

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Example: Applying the Power Rule for Exponents With Integer Bases

Express 5(5^2) \times 5^4 as a base raised to a single exponent.

EXPLANATION

First, we write 5(5^2) = 5^1 \times 5^2.

So our expression can be written as 5^1 \times 5^2 \times 5^4

To multiply powers of the same base, we simply add the exponents: 5^1 \times 5^2 \times 5^4 = 5^{1+2+4} = 5^7

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Practice: Applying the Power Rule for Exponents With Integer Bases

$ 7^6\times 7^3 = $

a
 $7^9$
b
 $7^3$
c
 $49^{18}$
d
 $14^9$
e
 $14^3$
Practice: Applying the Power Rule for Exponents With Integer Bases

$2^2\times 2 \left(2^3\right) = $

a
 $8^5$
b
 $2^6$
c
 $2^5$
d
 $4^5$
e
 $8^6$
Example: Applying the Product Rule for Exponents With Fractional or Decimal Bases

Express \left(\dfrac{1}{3}\right)^2\times \left(\dfrac{1}{3}\right)^2 as a base raised to a single exponent.

EXPLANATION

To multiply powers of the same base, we add the exponents: \left(\dfrac{1}{3}\right)^2\times \left(\dfrac{1}{3}\right)^2 = \left(\dfrac{1}{3}\right)^{2+2} = \left(\dfrac{1}{3}\right)^4

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Practice: Applying the Product Rule for Exponents With Fractional or Decimal Bases

$\left(\dfrac{1}{2}\right)^3\times \left(\dfrac{1}{2}\right)^3 = $

a
 $\left(\dfrac{1}{2}\right)^9$
b
 $\left(\dfrac{1}{2}\right)^{27}$
c
 $\left(\dfrac{1}{2}\right)^6$
d
 $\left(\dfrac{1}{4}\right)^6$
e
 $\left(\dfrac{1}{4}\right)^9$
Practice: Applying the Product Rule for Exponents With Fractional or Decimal Bases

$\left(-\dfrac{1}{4}\right)^9\times \left(-\dfrac{1}{4}\right)^4 = $

a
 $\left(\dfrac{1}{16}\right)^{13}$
b
 $\left(\dfrac{1}{16}\right)^{36}$
c
 $\left(-\dfrac{1}{2}\right)^{36}$
d
 $\left(-\dfrac{1}{4}\right)^{-5}$
e
 $\left(-\dfrac{1}{4}\right)^{13}$
The Product Rule With Negative Exponents

We can use the product rule for exponents with negative exponents too. For instance, to simplify 3^3\times 3^{-5}, we add the exponents: 3^{3} \times 3^{-5} = 3^{3-5} = 3^{-2}

Now, to evaluate 3^{-2}, we find the reciprocal of the base (3) and then raise that reciprocal to the same power but with the opposite sign:

3^{-2} = \left(\dfrac{1}{3}\right)^2 =\dfrac{1}{3} \times \dfrac{1}{3} = \dfrac{1}{9}

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Example: Evaluating Products with Negative Exponents Using the Product Rule

Evaluate 2^2\times 2^{-5}.

EXPLANATION

To multiply powers of the same base, we add the exponents: 2^2\times 2^{-5} =2^{2+(-5)} = 2^{-3}

To evaluate 2^{-3}, we find the reciprocal of the base (2) and then raise that reciprocal to the same power but with the opposite sign:

2^{-3} = \left(\dfrac{1}{2}\right)^3 =\dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{8}

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Practice: Evaluating Products with Negative Exponents Using the Product Rule

$5^{-3} \times 5^3 = $

a
 $1$
b
 $0$
c
 $5$
d
 $125$
e
 $\dfrac{1}{125}$
Practice: Evaluating Products with Negative Exponents Using the Product Rule

$ 5^{4}\times 5^{-1} = $

a
 $125$
b
 $15$
c
 $\dfrac 1{125}$
d
 $75$
e
 $\dfrac 1{75}$
Practice: Evaluating Products with Negative Exponents Using the Product Rule

$ 6^5\times 6^{-7} = $

a
 $-12$
b
 $\dfrac{1}{216}$
c
 $\dfrac{1}{36}$
d
 $\dfrac{1}{6}$
e
 $-18$
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