10.6 Laws of Exponents

In this section, we review the rules for performing operations on powers.

Product of Powers

Consider a product of two powers with the same base.

$(a^{3}) (a^{2}) = a a a \cdot a a = a^{5}$

because $a$ occurs as a factor five times. The number of $a$ 's in the product is the $s u m$ of the number of $a$ 's in each factor.

Here are some mistakes to avoid.

Caution

Note that we do not multiply the exponents when simplifying a product. For example,
$b^{4} \cdot b^{2} \neq b^{8}$
You can check this with your calculator by choosing a value for $b$ , for instance, $b equals 3$ :
$3^{4} \cdot 3^{2} \neq 3^{8}$
In order to apply the first law of exponents, the bases must be the same. For example,
$2^{3} \cdot 3^{5} \neq 6^{8}$
(Check this on your calculator.)
We do not multiply the bases when simplifying a product. In Examplea, note that
$5^{3} \cdot 5^{4} \neq 25^{7}$
Although we can simplify the product $x^{2} x^{3}$ as $x to the power 5$ , we cannot simplify the sum $x^{2} + x^{3}$ , because $x^{2}$ and $x^{3}$ are not like terms.

Example 2

Multiply $(- 3 x^{4} z^{2}) (5 x^{3} z)$ .

Rearrange the factors to group the numerical coefficients and the powers of each base. Apply the first law of exponents.

$\begin{aligned} (- 3 x^{4} z^{2}) (5 x^{3} z) & = (- 3) (5) x^{4} x^{3} z^{2} z \\ = - 15 x^{7} z^{3} \end{aligned}$

Quotients of Powers

To reduce a fraction, we divide both numerator and denominator by any common factors.

$\frac{x^{7}}{x^{4}} = \frac{x x x x x x x}{x x x x} = \frac{x^{3}}{1} = x^{3}$

We can obtain the same result more quickly by subtracting the exponent of the denominator from the exponent of the numerator.

$\frac{x^{7}}{x^{4}} = x^{7 - 4} = x^{3}$

What if the larger power occurs in the denominator of the fraction?

$\frac{x^{4}}{x^{7}} = \frac{x x x x}{x x x x x x x} = \frac{1}{x^{3}}$

In this case, we subtract the exponent of the numerator from the exponent of the denominator.

$\frac{x^{4}}{x^{7}} = \frac{1}{x^{7 - 4}} = \frac{1}{x^{3}}$

These examples suggest the following law.

Note

Second Law of Exponents: Quotient of Powers

To divide two powers with the same base, subtract the smaller exponent from the larger one, keeping the same base.

If the larger exponent occurs in the numerator, put the power in the numerator.
$If m > n, then \frac{a^{m}}{a^{n}} = a^{m - n} (a \neq 0)$
If the larger exponent occurs in the denominator, put the power in the denominator.
$If m < n, then \frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}} (a \neq 0)$

Example 3

$\frac{3^{8}}{3^{2}} = 3^{8 - 2} = 3^{6} Subtract exponents: 8 > 2.$
$\frac{w^{3}}{w^{6}} = \frac{1}{w^{6 - 3}} = \frac{1}{w^{3}} Subtract exponents: 3 < 6.$

Example 4

Divide $\frac{3 x^{2} y^{4}}{6 x^{3} y}$

Consider the numerical coefficients and the powers of each variable separately. Use the second law of exponents to simplify each quotient of powers.

$\begin{aligned} \frac{3 x^{2} y^{4}}{6 x^{3} y} & = \frac{3}{6} \cdot \frac{x^{2}}{x^{3}} \cdot \frac{y^{4}}{y} & Subtract exponents. \\ = \frac{1}{2} \cdot \frac{1}{x^{3 - 2}} \cdot y^{4 - 1} \\ = \frac{1}{2} \cdot \frac{1}{x} \cdot y^{3} = \frac{y^{3}}{2 x} \end{aligned}$

Power of a Power

Consider the expression $open parenthesis a to the power 4 close parenthesis cubed$ , the third power of $a to the power 4$ .

${(a^{4})}^{3} = (a^{4}) (a^{4}) (a^{4}) = a^{4 + 4 + 4} = a^{12} Add exponents.$

We can obtain the same result by multiplying the exponents together.

${(a^{4})}^{3} = a^{4 \cdot 3} = a^{12}$

Power of a Product

To simplify the expression $open parenthesis 5 a close parenthesis cubed$ , we use the associative and commutative laws to regroup the factors as follows.

$\begin{aligned} (5 a)^{3} & = (5 a) (5 a) (5 a) \\ = 5 \cdot 5 \cdot 5 \cdot a \cdot a \cdot a \\ = 5^{3} a^{3} \end{aligned}$

Thus, to raise a product to a power, we can simply raise each factor to the power.

Example 6

$(5 a)^{3} = 5^{3} a^{3} = 125 a^{3} Cube each factor$ .
$\begin{aligned} {(- x y^{2})}^{4} & = (- x)^{4} {(y^{2})}^{4} & Raise each factor to the fourth power. \\ = x^{4} y^{8} & Apply the third law of exponents. \end{aligned}$

Caution

Compare the two expressions $3 a squared$ and $open parenthesis 3 a close parenthesis squared$ ; they are not the same. In the expression $3 a squared$ , only the factor $a$ is squared. But in $open parenthesis 3 a close parenthesis squared$ , both $3$ and $a$ are squared. Thus,
$3 a squared cannot be simplified$
but
$(3 a)^{2} = 3^{2} a^{2} = 9 a^{2}$
Compare the two expressions $open parenthesis 3 a close parenthesis squared$ and $open parenthesis 3 plus a close parenthesis squared$ . The fourth law of exponents applies to the product $3 a$ , but not to the sum $3 plus a$ . Thus,
$(3 + a)^{2} \neq 3^{2} + a^{2}$
In order to simplify $open parenthesis 3 plus a close parenthesis squared$ , we must expand the binomial product:
$(3 + a)^{2} = (3 + a) (3 + a) = 9 + 6 a + a^{2}$

Power of a Quotient

To simplify the expression ${(\frac{x}{3})}^{4}$ , we multiply together $4$ copies of the fraction $\frac{x}{3}$ .

$\begin{aligned} {(\frac{x}{3})}^{4} & = \frac{x}{3} \cdot \frac{x}{3} \cdot \frac{x}{3} \cdot \frac{x}{3} = \frac{x \cdot x \cdot x \cdot x}{3 \cdot 3 \cdot 3 \cdot 3} \\ = \frac{x^{4}}{3^{4}} = \frac{x^{4}}{81} \end{aligned}$

In general, we have the following rule.

For reference, we state all of the laws of exponents together. All the laws are valid when a and b are not equal to zero and when the exponents m and n are whole numbers.

Note

Laws of Exponents

$a^{m} \cdot a^{n} = a^{m + n}$
1. $\frac{a^{m}}{a^{n}} = a^{m - n} m > n$
2. $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}} m < n$
${(a^{m})}^{n} = a^{m + n}$
$(a b)^{n} = a^{n} b^{n}$
${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

Example 7

Simplify $5 x squared y cubed open parenthesis 2 x y squared close parenthesis to the power 4$

According to the order of operations, we should perform any powers before multiplications. Thus, we begin by simplifying $open parenthesis 2 x y squared close parenthesis to the power 4$ . We apply the fourth law.

$\begin{aligned} 5 x^{2} y^{3} {(2 x y^{2})}^{4} & = 5 x^{2} y^{3} \cdot 2^{4} x^{4} {(y^{2})}^{4} & Apply the fourth law. \\ = 5 x^{2} y^{3} \cdot 2^{4} x^{4} y^{8} \end{aligned}$

Finally, multiply powers with the same base. Apply the first law.

$5 x^{2} y^{3} \cdot 2^{4} x^{4} y^{8} = 5 \cdot 2^{4} x^{2} x^{4} y^{3} y^{8} = 80 x^{6} y^{11}$

Example 8

Simplify ${(\frac{2 x}{z^{2}})}^{3}$

Begin by applying the fifth law.

$\begin{aligned} {(\frac{2 x}{z^{2}})}^{3} & = \frac{(2 x)^{3}}{{(z^{2})}^{2}} & Apply the fourth law to the numerator and the third law to the denominator. \\ = \frac{2^{3} x^{3}}{z^{6}} = \frac{8 x^{3}}{z^{6}} \end{aligned}$

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Exponent
Power

SKILLS

Practice each skill in the exercises listed.

Apply the laws of exponents: #1–8
Simplify expressions: #9–16, 25–32
Multiply and divide power: #17–24

Exercises A.6

For Problems 1–8, simplify by applying the appropriate law of exponents.

$b^{4} \cdot b^{5}$
$b^{2} \cdot b^{8}$
$open parenthesis q cubed close parenthesis open parenthesis q close parenthesis open parenthesis q to the power 5 close parenthesis$
$(p^{2}) (p^{4}) (p^{4})$

$b to the power 9$
$b to the power 10$
$q to the power 9$
$p to the power 10$

$\frac{w^{6}}{w^{3}}$
$\frac{c^{12}}{c^{4}}$
$\frac{z^{6}}{z^{9}}$
$\frac{b^{4}}{b^{8}}$

$2^{7} \cdot 2^{2}$
$6^{5} \cdot 6^{3}$
$\frac{2^{9}}{2^{4}}$
$\frac{8^{6}}{8^{2}}$

$2 to the power 9$
$6 to the power 8$
$2 to the power 5$
$8 to the power 4$

$open parenthesis d cubed close parenthesis to the power 5$
$open parenthesis d to the power 4 close parenthesis squared$
$open parenthesis 5 to the power 4 close parenthesis cubed$
$open parenthesis 4 cubed close parenthesis cubed$

$open parenthesis 6 x close parenthesis cubed$
$open parenthesis 3 y close parenthesis to the power 4$
$open parenthesis 2 t cubed close parenthesis to the power 5$
$open parenthesis 6 s squared close parenthesis squared$

$216 x cubed$
$81 y to the power 4$
$32 t to the power 15$
$36 s to the power 4$

${(\frac{w}{2})}^{6}$
${(\frac{5}{u})}^{4}$
${(\frac{- 4}{p^{5}})}^{3}$
${(\frac{- 3}{q^{4}})}^{5}$

${(\frac{h^{2}}{m^{3}})}^{4}$
${(\frac{n^{3}}{k^{4}})}^{8}$
$(- 4 a^{2} b^{4})^{4}$
$(- 5 a b^{8})^{3}$

$\frac{h^{8}}{m^{12}}$
$\frac{n^{24}}{k^{32}}$
$2565 a^{8} b^{16}$
$- 125 a^{3} b^{24}$

$\frac{a b^{2}}{(a b)^{2}}$
$\frac{(x^{2} y)^{2}}{x^{2} y^{2}}$
$\frac{(2 m p)^{3}}{2 m^{3} p}$
$\frac{4^{2} r t^{4}}{2^{4} r^{4} t}$

For Problems 9–15, simplify if possible.

$w + w$
$w (w)$

$2 w$
$w^{2}$

$m^{2} - m^{2}$
$m^{2} (- m^{2})$

$4 z squared minus 6 z squared$
$4 z squared open parenthesis minus 6 z squared close parenthesis$

$minus 2 z squared$
$m - 24 z^{4}$

$t cubed plus 3 t cubed$
$t cubed open parenthesis 3 t cubed close parenthesis$

$4 p squared plus 3 p cubed$
$4 p squared open parenthesis 3 p cubed close parenthesis$

Cannot be simplified
$12 p to the power 5$

$2 w^{2} - 5 w^{4}$
$(2 w^{2}) (- 5 w^{4})$

$3^{9} \cdot 3^{8}$
$3^{9} + 3^{8}$

$3 to the power 17$
Cannot be simplified

$(- 2)^{7} (- 2)^{5}$
$- 2^{7} - 2^{5}$

F mor Problems 17–20, multiply.

$open parenthesis 4 y close parenthesis open parenthesis minus 6 y close parenthesis$
$(- 4 z) (- 8 z)$

$minus 24 y squared$
$32 z squared$

$open parenthesis 2 w z cubed close parenthesis open parenthesis minus 8 z close parenthesis$
$open parenthesis 4 w z close parenthesis open parenthesis minus 9 w squared z squared close parenthesis$

$minus 4 x open parenthesis 3 x y close parenthesis open parenthesis x y cubed close parenthesis$
$open parenthesis minus 5 x squared close parenthesis open parenthesis 2 x y close parenthesis open parenthesis 5 x squared close parenthesis$

$- 12 x^{3} y^{4}$
$- 50 x^{5} y$

$- 7 a b^{2} (- 3 a b^{3})$
$- 4 a^{2} b (- 3 a^{3} b^{2})$

F dor Problems 21–22, divide.

$\frac{2 a^{3} b}{8 a^{4} b^{5}}$
$\frac{8 a^{2} b}{12 a^{5} b^{3}}$

$\frac{1}{4 a b^{4}}$
$\frac{2}{3 a^{3} b^{2}}$

$\frac{- 12 q w^{4}}{8 q w^{2}}$
$\frac{- 12 r z^{6}}{20 r z}$

For Problems 23–24, multiply or divide.

$\frac{- 15 b c (b^{2} c)}{- 3 b^{3} c^{4}}$
$\frac{- 25 c (c^{2} d^{2})}{- 5 c^{8} d^{2}}$

$\frac{5}{c^{2}}$
$\frac{5}{c^{5}}$

$- 2 x^{3} (x^{2} y) (- 4 y^{2})$
$3 x y^{3} (- x^{4}) (- 2 y^{2})$

For Problems 25–28, simplify by applying the laws of exponents.

$b cubed open parenthesis b squared close parenthesis to the power 5$
$b (b^{4})^{6}$

$b to the power 13$
$b to the power 25$

$(p^{2} q)^{3} (p q^{3})$
$(p^{3})^{4} (p^{3} q^{4})$

$open parenthesis 2 x cubed y close parenthesis squared open parenthesis x y cubed close parenthesis to the power 4$
$open parenthesis 3 x y squared close parenthesis cubed open parenthesis 2 x squared y squared close parenthesis squared$

$4 x^{10} y^{14}$
$108 x^{7} y^{10}$

$- a^{2} (- a)^{2}$
$- a^{3} (- a)^{3}$

For Problems 29–32, simplify by applying the laws of exponents.

${(\frac{- 2 x}{3 y^{2}})}^{3}$
${(\frac{- x^{2}}{2 y})}^{4}$

$\frac{- 8 x^{3}}{27 y^{6}}$
$\frac{x^{8}}{16 y^{4}}$

$\frac{(4 x)^{3}}{(- 2 x^{2})^{2}}$
$\frac{(5 x)^{2}}{(- 3 x^{2})^{3}}$

$\frac{(x y)^{2} (- x^{2} y)^{3}}{(x^{2} y^{2})^{2}}$
$\frac{(- x^{2}) (- x^{2})^{4}}{(x^{2})^{3}}$

$- x^{4} y$
$x to the power 4$

$(\frac{- 2 x}{y^{2}}) {(\frac{y^{2}}{3 x})}^{2}$
${(\frac{x^{2} z}{2})}^{3} {(\frac{- 2}{x^{2} z})}^{3}$

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.