3.7: Negative Exponents

Overview

• Reciprocals
• Negative Exponents
• Working with Negative Exponents

Reciprocals

Reciprocals

Two real numbers are said to be reciprocals of each other if their product is 1. Every nonzero real number has exactly one reciprocal, as shown in the examples below. Zero has no reciprocal.

Example $\PageIndex{1}$

$4 \cdot \dfrac{1}{4} = 1$. This means that $4$ and $\dfrac{1}{4}$ are reciprocals.

Example $\PageIndex{2}$

$6 \cdot \dfrac{1}{6} = 1$. This means that $6$ and $\dfrac{1}{6}$ are reciprocals.

Example $\PageIndex{3}$

$-2 \cdot \dfrac{-1}{2} = 1$. This means that $-2$ and $-\dfrac{1}{2}$ are reciprocals.

Example $\PageIndex{4}$

$a \cdot \dfrac{1}{a} = 1$. This means that $a$ and $\dfrac{1}{a}$ are reciprocals.

Example $\PageIndex{5}$

$x \cdot \dfrac{1}{x} = 1$. This means that $x$ and $\dfrac{1}{x}$ are reciprocals.

Example $\PageIndex{6}$

$x^3 \cdot \dfrac{1}{x^3} = 1$. This means that $x^3$ and $\dfrac{1}{x^3}$ are reciprocals.

Negative Exponents

We can use the idea of reciprocals to find a meaning for negative exponents.

Consider the product of $x^3$ and $x^{-3}$. Assume $x \not = 0$.

$$x^3 \cdot x^{-3} = x^{3 + (-3)} = x^0 = 1$$

Thus, since the product of $x^3$ and $x^{-3}$ is $1$, $x^3$ and $x^{-3}$ must be reciprocals.

We also know that $x^3 \cdot \dfrac{1}{x^3} = 1$. (See problem 6 above.) Thus, $x^3$ and $\dfrac{1}{x^3}$ are also reciprocals.

Then, since $x^{-3}$ and $\dfrac{1}{x^3}$ are both reciprocals of $x^3$ and a real number can have only one reciprocal, it must be that $x^{-3} = \dfrac{1}{x^3}$.

We have used $-3$ as the exponent, but the process works as well for all other negative integers. We make the following definition.:

Negative Exponent Definition

If $n$ is any natural number and $x$ is any nonzero real number, then:

$x^{-n} = \dfrac{1}{x^n}$

Sample Set A

Write each of the following so that only positive exponents appear.

Example $\PageIndex{7}$

$x^{-6} = \dfrac{1}{x^6}$

Example $\PageIndex{8}$

$a^{-1} = \dfrac{1}{a^a} = \dfrac{1}{a}$

Example $\PageIndex{9}$

$7^{-2} = \dfrac{1}{7^2} = \dfrac{1}{49}$

Example $\PageIndex{10}$

$(3a)^{-6} = \dfrac{1}{(3a)^6}$

Example $\PageIndex{11}$

$(5x-1)^{-24} = \dfrac{1}{(5x-1)^{-24}}$

Example $\PageIndex{12}$

$(k+2x)^{-(-8)} = (k+2z)^8$

Practice Set A

Write each of the following using only positive exponents.

Practice Problem $\PageIndex{1}$

$y^{-5}$

Answer

$\dfrac{1}{y^5}$

Practice Problem $\PageIndex{2}$

$m^{-2}$

Answer

$\dfrac{1}{m^2}$

Practice Problem $\PageIndex{3}$

$3^{-2}$

Answer

$\dfrac{1}{9}$

Practice Problem $\PageIndex{4}$

$5^{-1}$

Answer

$\dfrac{1}{5}$

Practice Problem $\PageIndex{5}$

$2^{-4}$

Answer

$\dfrac{1}{16}$

Practice Problem $\PageIndex{6}$

$(xy)^{-4}$

Answer

$\dfrac{1}{(xy)^4}$

Practice Problem $\PageIndex{7}$

$(a+2b)^{-12}$

Answer

$\dfrac{1}{(a+2b)^{12}}$

Practice Problem $\PageIndex{8}$

$(m-n)^{-(-4)}$

Answer

$(m-n)^4$

Note

It is important to note that $a^{-n}$ is not necessarily a negative number. For example,

$3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ $3^{-2} \not = \ -9$

Working with Negative Exponents

The problems of Sample Set A suggest the following rule for working with exponents:

Moving Factors Up and Down

In a fraction, a factor can be moved from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent.

Sample Set B

Write each of the following so that only positive exponents appear.

Example $\PageIndex{13}$

$x^{-2}y^5$.

The factor $x^{-2}$ can be moved from the numerator to the denominator by changing the exponent $-2$ to $+2$

$x^{-2}y^5 = \dfrac{y^5}{x^2}$

Example $\PageIndex{14}$

$a^9b^{-3}$.

The factor $b^{-3}$ can be moved from the numerator to the denominator by changing the exponent $-3$ to $+3$.

$a^9b^{-3} = \dfrac{a^9}{b^3}$

Example $\PageIndex{15}$

$\dfrac{a^4b^2}{c^{-6}}$.

This fraction can be written without any negative exponents by moving the factor $c^{-6}$ into the numerator.
We must change the $-6$ to $+6$ to make the move legitimate.

$\dfrac{a^4b^2}{c^{-6}} = a^4b^2c^6$

Example $\PageIndex{16}$

$\dfrac{1}{x^{-3}y^{-2}z^{-1}}$.

This fraction can be written without negative exponents by moving all the factors from the denominator to the numerator. Change the sign of each exponent: $-3$ to $+3$, $-2$ to $+2$, $-1$ to $+1$.

$\dfrac{1}{x^{-3}y^{-2}z^{-1}} = x^3y^2z^1 = x^3y^2z$

Practice Set B

Write each of the following so that only positive exponents appear.

Practice Problem $\PageIndex{9}$

$x^{-4}y^7$

Answer

$\dfrac{y^7}{x^4}$

Practice Problem $\PageIndex{10}$

$\dfrac{a^2}{b^{-4}}$

Answer

$a^2b^4$

Practice Problem $\PageIndex{11}$

$\dfrac{x^3y^4}{z^{-8}}$

Answer

$x^3y^4z^8$

Practice Problem $\PageIndex{12}$

$\dfrac{6m^{-3}n^{-2}}{7k^{-1}}$

Answer

$\dfrac{6k}{7m^3n^2}$

Practice Problem $\PageIndex{13}$

$\dfrac{1}{a^{-2}b^{-6}c^{-8}}$

Answer

$a^2b^6c^8$

Practice Problem $\PageIndex{14}$

$\dfrac{3a(a-5b)^{-2}}{5b(a-4b)^5}$

Answer

$\dfrac{3a}{5b(a-5b)^2(a-4b)^5}$

Sample Set C

Example $\PageIndex{17}$

Rewrite $\dfrac{24 a^{7} b^{9}}{2^{3} a^{4} b^{-6}}$ in a simpler form.
Notice that we are dividing powers with the same base. We'll proceed by using the rules of exponents.
$\begin{aligned} \dfrac{24 a^{7} b^{9}}{2^{3} a^{4} b^{-6}}=\dfrac{24 a^{7} b^{9}}{8 a^{4} b^{-6}} &=3 a^{7-4} b^{9-(-6)} \\ &=3 a^{3} b^{9+6} \\ &=3 a^{3} b^{15} \end{aligned}$

Example $\PageIndex{17}$

Write $\dfrac{9 a^{5} b^{3}}{5 x^{3} y^{2}}$ so that no denominator appears.
We can eliminate the denominator by moving all factors that make up the denominator to the numerator.
$9 a^{5} b^{3} 5^{-1} x^{-3} y^{-2}$

Example $\PageIndex{17}$

Find the value of $\dfrac{1}{10^{-2}}+\dfrac{3}{4^{-3}}$
We can evaluate this expression by eliminating the negative exponents.
$\begin{aligned} \dfrac{1}{10^{-2}}+\dfrac{3}{4^{-3}} &=1 \cdot 10^{2}+3 \cdot 4^{3} \\ &=1 \cdot 100+3 \cdot 64 \\ &=100+192 \\ &=292 \end{aligned}$

Practice Set C

Practice Problem $\PageIndex{15}$

Rewrite $\dfrac{36x^8b^3}{3^2x^{-2}b^{-5}}$ in a simpler form.

Answer

$4x^{10}b^8$

Practice Problem $\PageIndex{16}$

Write $\dfrac{2^4m^{-3}n^7}{4^{-1}x^5}$ in a simpler form and one in which no denominator appears.

Answer

$64m^{-3}n^7x^{-5}$

Practice Problem $\PageIndex{17}$

Find the value of $\dfrac{2}{5^{-2}} + 6^{-2} \cdot 2^3 \cdot 3^2$

Answer

$52$

Exercises

Write the following expressions using only positive exponents. Assume all variables are nonzero.

Exercise $\PageIndex{1}$

$x^{-2}$

Answer

$\dfrac{1}{x^{-2}}$

Exercise $\PageIndex{2}$

$x^{-4}$

Exercise $\PageIndex{3}$

$x^{-7}$

Answer

$\dfrac{1}{x^7}$

Exercise $\PageIndex{4}$

$a^{-8}$

Exercise $\PageIndex{5}$

$a^{-10}$

Answer

$\dfrac{1}{a^{-10}}$

Exercise $\PageIndex{6}$

$b^{-12}$

Exercise $\PageIndex{7}$

$b^{-14}$

Answer

$\dfrac{1}{b^{14}}$

Exercise $\PageIndex{8}$

$y^{-1}$

Exercise $\PageIndex{9}$

$y^{-5}$

Answer

$\dfrac{1}{y^5}$

Exercise $\PageIndex{10}$

$(x+1)^{-2}$

Exercise $\PageIndex{11}$

$(x-5)^{-3}$

Answer

$\dfrac{1}{(x-5)^3}$

Exercise $\PageIndex{12}$

$(y-4)^{-6}$

Exercise $\PageIndex{13}$

$(a+9)^{-10}$

Answer

$\dfrac{1}{(a+9)^{10}}$

Exercise $\PageIndex{14}$

$(r+3)^{-8}$

Exercise $\PageIndex{15}$

$(a-1)^{-12}$

Answer

$\dfrac{1}{(a-1)^{12}}$

Exercise $\PageIndex{16}$

$x^3y^{-2}$

Exercise $\PageIndex{17}$

$x^7y^{-5}$

Answer

$\dfrac{x^7}{y^5}$

Exercise $\PageIndex{18}$

$a^4b^{-1}$

Exercise $\PageIndex{19}$

$a^7b^{-8}$

Answer

$\dfrac{a^7}{b^8}$

Exercise $\PageIndex{20}$

$a^2b^3c^{-2}$

Exercise $\PageIndex{21}$

$x^3y^2z^{-6}$

Answer

$\dfrac{x^3y^2}{z^6}$

Exercise $\PageIndex{22}$

$x^3y^{-4}z^2w$

Exercise $\PageIndex{23}$

$a^7b^{-9}zw^3$

Answer

$\dfrac{a^7zw^3}{b^9}$

Exercise $\PageIndex{24}$

$a^3b^{-1}zw^2$

Exercise $\PageIndex{25}$

$x^5y^{-5}z^{-2}$

Answer

$\dfrac{x^5}{y^5z^2}$

Exercise $\PageIndex{26}$

$x^4y^{-8}z^{-3}w^{-4}$

Exercise $\PageIndex{27}$

$a^{-4}b^{-6}c^{-1}d^4$

Answer

$\dfrac{d^4}{a^4b^6c}$

Exercise $\PageIndex{28}$

$x^9y^{-6}z^{-1}w^{-5}r^{-2}$

Exercise $\PageIndex{29}$

$4x^{-6}y^2$

Answer

$\dfrac{4y^2}{x^6}$

Exercise $\PageIndex{30}$

$5x^2y^2z^{-5}$

Exercise $\PageIndex{31}$

$7a^{-2}b^2c^2$

Answer

$\dfrac{7b^2c^2}{a^2}$

Exercise $\PageIndex{32}$

$4x^3(x+1)^2y^{-4}z^{-1}$

Exercise $\PageIndex{33}$

$7a^2(a-4)^3b^{-6}c^{-7}$

Answer

$\dfrac{7a^2(a-4)^3}{b^6c^7}$

Exercise $\PageIndex{34}$

$18b^{-6}(b^2-3)^{-5}c^{-4}d^5e^{-1}$

Exercise $\PageIndex{35}$

$7(w+2)^{-2}(w+1)^3$

Answer

$\dfrac{7(w+1)^3}{(w+2)^2}$

Exercise $\PageIndex{36}$

$2(a-8)^{-3}(a-2)^5$

Exercise $\PageIndex{37}$

$(x^2+3)^3(x^2-1)^{-4}$

Answer

$\dfrac{(x^2+3)^3}{(x^2-1)^4}$

Exercise $\PageIndex{38}$

$(x^4+2x-1)^{-6}(x+5)^4$

Exercise $\PageIndex{39}$

$(3x^2-4x-8)^{-9}(2x+11)^{-3}$

Answer

$\dfrac{1}{(3x^2-4x-8)^{9}(2x+11)^{3}}$

Exercise $\PageIndex{40}$

$(5y^2+8y-6)^{-2}(6y-1)^{-7}$

Exercise $\PageIndex{41}$

$7a(a^2-4)^{-2}(b^2-1)^{-2}$

Answer

$\dfrac{7a}{(a^2-4)^2(b^2-1)^2}$

Exercise $\PageIndex{42}$

$(x-5)^{-4}3b^2c^4(x+6)^8$

Exercise $\PageIndex{43}$

$(y^3+1)^{-1}5y^3z^{-4}w^{-2}(y^3-1)^{-2}$

Answer

$\dfrac{5y^3}{(y^3+1)z^4w^2(y^3-1)^2}$

Exercise $\PageIndex{44}$

$5x^3(2x^{-7})$

Exercise $\PageIndex{45}$

$3y^{-3}(9x)$

Answer

$\dfrac{27x}{y^3}$

Exercise $\PageIndex{46}$

$6a^{-4}(2a^{-6})$

Exercise $\PageIndex{47}$

$4a^2b^2a^{-5}b^{-2}$

Answer

$\dfrac{4}{a^3}$

Exercise $\PageIndex{48}$

$5^{-1}a^{-2}b^{-6}b^{-11}c^{-3}c^9$

Exercise $\PageIndex{49}$

$2^3x^22^{-3}x^{-2}$

Answer

$1$

Exercise $\PageIndex{50}$

$7a^{-3}b^{-9} \cdot 5a^6bc^{-2}c^4$

Exercise $\PageIndex{51}$

$(x+5)^2(x+5)^{-6}$

Answer

$\dfrac{1}{(x+5)^4}$

Exercise $\PageIndex{52}$

$(a-4)^3(a-4)^{-10}$

Exercise $\PageIndex{53}$

$8(b+2)^{-8}(b+2)^{-4}(b+2)^3$

Answer

$\dfrac{8}{(b+2)^9}$

Exercise $\PageIndex{54}$

$3a^5b^{-7}(a^2+4)^{-3}6a^{-4}b(a^2+4)^{-1}(a^2+4)$

Exercise $\PageIndex{55}$

$-4a^3b^{-5}(2a^2b^7c^{-2})$

Answer

$\dfrac{-8a^5b^2}{c^2}$

Exercise $\PageIndex{56}$

$-2x^{-2}y^{-4}z^4(-6x^3y^{-3}z)$

Exercise $\PageIndex{57}$

$(-5)^2(-5)^{-1}$

Answer

$-5$

Exercise $\PageIndex{58}$

$(-9)^{-3}(9)^3$

Exercise $\PageIndex{59}$

$(-1)^{-1}(-1)^{-1}$

Answer

$1$

Exercise $\PageIndex{60}$

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

Exercise $\PageIndex{61}$

$\dfrac{1}{a^{-4}}$

Answer

$a^4$

Exercise $\PageIndex{62}$

$\dfrac{1}{a^{-1}}$

Exercise $\PageIndex{63}$

$\dfrac{4}{x^{-6}}$

Answer

$4x^6$

Exercise $\PageIndex{64}$

$\dfrac{7}{x^{-8}}$

Exercise $\PageIndex{65}$

$\dfrac{23}{y^{-1}}$

Answer

$23y$

Exercise $\PageIndex{66}$

$\dfrac{6}{a^2b^{-4}}$

Exercise $\PageIndex{67}$

$\dfrac{3c^5}{a^3b^{-3}}$

Answer

$\dfrac{3b^3c^5}{a^3}$

Exercise $\PageIndex{68}$

$\dfrac{16a^{-2}b^{-6}c}{2yz^{-5}w^{-4}}$

Exercise $\PageIndex{69}$

$\dfrac{24y^2z^{-8}}{6a^2b^{-1}c^{-9}d^3}$

Answer

$\dfrac{4bc^9y^2}{a^2d^3z^8}$

Exercise $\PageIndex{70}$

$\dfrac{3^{-1}b^5(b+7)^{-4}}{9^{-1}a^{-4}(a+7)^2}$

Exercise $\PageIndex{71}$

$\dfrac{36a^6b^5c^8}{3^2a^3b^7c^9}$

Answer

$\dfrac{4a^3}{b^2c}$

Exercise $\PageIndex{72}$

$\dfrac{45a^4b^2c^6}{15a^2b^7c^8}$

Exercise $\PageIndex{73}$

$\dfrac{3^3x^4y^3z}{3^2xy^5z^5}$

Answer

$\dfrac{3x^3}{y^2z^4}$

Exercise $\PageIndex{74}$

$\dfrac{21x^2y^2z^5w^4}{7xyz^{12}w^{14}}$

Exercise $\PageIndex{75}$

$\dfrac{33a^{-4}b^{-7}}{11a^3b^{-2}}$

Answer

$\dfrac{3}{a^7b^5}$

Exercise $\PageIndex{76}$

$\dfrac{51x^{-5}y^{-3}}{3xy}$

Exercise $\PageIndex{77}$

$\dfrac{2^6x^{-5}y^{-2}a^{-7}b^5}{2^{-1}x^{-4}y^{-2}b^6}$

Answer

$\dfrac{128}{a^7bx}$

Exercise $\PageIndex{78}$

$\dfrac{(x+3)^3(y-6)^4}{(x+3)^5(y-6)^{-8}}$

Exercise $\PageIndex{79}$

$\dfrac{4x^3}{y^7}$

Answer

$\dfrac{4x^3}{y^7}$

Exercise $\PageIndex{80}$

$\dfrac{5x^4y^3}{a^3}$

Exercise $\PageIndex{81}$

$\dfrac{23a^4b^5c^{-2}}{x^{-6}y^5}$

Answer

$\dfrac{23a^4b^5x^6}{c^2y^5}$

Exercise $\PageIndex{82}$

$\dfrac{2^3b^5c^2d^{-9}}{4b^4cx}$

Exercise $\PageIndex{83}$

$\dfrac{10x^3y^{-7}}{3x^5z^2}$

Answer

$\dfrac{10}{3x^2y^7z^2}$

Exercise $\PageIndex{84}$

$\dfrac{3x^2y^{-2}(x-5)}{9^{-1}(x+5)^3}$

Exercise $\PageIndex{85}$

$\dfrac{14a^2b^2c^{-12}(a^2+21)^{-4}}{4^{-2}a^2b^{-1}(a+6)^3}$

Answer

$\dfrac{224b^3}{c^{12}(a^2+21)^4(a+6)^3}$

For the following problems, evaluate each numerical expression.

Exercise $\PageIndex{86}$

$4^{-1}$

Exercise $\PageIndex{87}$

$7^{-1}$

Answer

$\dfrac{1}{7}$

Exercise $\PageIndex{88}$

$6^{-2}$

Exercise $\PageIndex{89}$

$2^{-5}$

Answer

$\dfrac{1}{32}$

Exercise $\PageIndex{90}$

$3^{-4}$

Exercise $\PageIndex{91}$

$6 \cdot 3^{-3}$

Answer

$\dfrac{2}{9}$

Exercise $\PageIndex{92}$

$4 \cdot 9^{-2}$

Exercise $\PageIndex{93}$

$28 \cdot 14^{-1}$

Answer

$2$

Exercise $\PageIndex{94}$

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

Exercise $\PageIndex{95}$

$2^{-1} \cdot 3^{-1} \cdot 4^{-1}$

Answer

$\dfrac{1}{24}$

Exercise $\PageIndex{96}$

$10^{-2} + 3(10^{-2})$

Exercise $\PageIndex{97}$

$(-3)^{-2}$

Answer

$\dfrac{1}{9}$

Exercise $\PageIndex{98}$

$(-10)^{-1}$

Exercise $\PageIndex{99}$

$\dfrac{3}{2^{-3}}$

Answer

$24$

Exercise $\PageIndex{100}$

$\dfrac{4^{-1}}{5^{-2}}$

Exercise $\PageIndex{101}$

$\dfrac{2^4-7}{4^{-1}}$

Answer

$36$

Exercise $\PageIndex{102}$

$\dfrac{2^{-1}+4^{-1}}{2^{-2} + 4^{-2}}$

Exercise $\PageIndex{103}$

$\dfrac{21^0-2^6}{2 \cdot 6-13}$

Answer

$63$

For the following problems, write each expression so that only positive exponents appear.

Exercise $\PageIndex{104}$

$(a^6)^{-2}$

Exercise $\PageIndex{105}$

$(a^5)^{-3}$

Answer

$\dfrac{1}{a^{15}}$

Exercise $\PageIndex{106}$

$(x^7)^{-4}$

Exercise $\PageIndex{107}$

$(x^4)^{-8}$

Answer

$\dfrac{1}{x^{32}}$

Exercise $\PageIndex{108}$

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

Exercise $\PageIndex{109}$

$(b^{-4})^{-1}$

Answer

$b^4$

Exercise $\PageIndex{110}$

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

Exercise $\PageIndex{111}$

$(y^{-9})^{-3}$

Answer

$y^{27}$

Exercise $\PageIndex{112}$

$(a^{-1})^{-1}$

Exercise $\PageIndex{113}$

$(b^{-1})^{-1}$

Answer

$b$

Exercise $\PageIndex{114}$

$(a^0)^{-1}$, $a \not = 0$

Exercise $\PageIndex{115}$

$(m^))^{-1}$, $m \not = 0$

Answer

$1$

Exercise $\PageIndex{116}$

$(x^{-3}y^7)^{-4}$

Exercise $\PageIndex{117}$

$(x^6y^6z^{-1})^2$

Answer

$\dfrac{x^{12}y^{12}}{z^2}$

Exercise $\PageIndex{118}$

$(a^{-5}b^{-1}c^0)^6$

Exercise $\PageIndex{119}$

$(\dfrac{y^3}{x^{-4}})^5$

Answer

$x^{20}y^{15}$

Exercise $\PageIndex{120}$

$(\dfrac{a^{-8}}{b^{-6}})^3$

Exercise $\PageIndex{121}$

$(\dfrac{2a}{b^3})^4$

Answer

$\dfrac{16a^4}{b^{12}}$

Exercise $\PageIndex{122}$

$(\dfrac{3b}{a^2})^{-5}$

Exercise $\PageIndex{123}$

$(\dfrac{5^{-1}a^3b^{-6}}{x^{-2}y^9})^2$

Answer

$\dfrac{a^6x^4}{25b^{12}y^{18}}$

Exercise $\PageIndex{124}$

$(\dfrac{4m^{-3}n^6}{2m^{-5}n})^3$

Exercise $\PageIndex{125}$

$(\dfrac{r^5s^{-4}}{m^{-8}n^7})^{-4}$

Answer

$\dfrac{n^{28}s^{16}}{m^{32}r^{20}}$

Exercise $\PageIndex{126}$

$(\dfrac{h^{-2}j^{-6}}{k^{-4}p})^{-5}$

Exercises for Review

Exercise $\PageIndex{127}$

Simplify $(4x^5y^3z^0)^3$

Answer

$64x^{15}y^9$

Exercise $\PageIndex{128}$

Find the sum. $-15 + 3$

Exercise $\PageIndex{129}$

Find the difference. $8 -(-12)$

Answer

$20$

Exercise $\PageIndex{130}$

Simplify $(-3)(-8) + 4(-5)$

Exercise $\PageIndex{131}$

Find the value of $m$ if $m = \dfrac{-3k-5t}{kt+6}$ when $k = 4$ and $t = -2$

Answer

$1$