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Simplify: ⓐ $8{p}^{\mathrm{-1}}$ ⓑ ${\left(8p\right)}^{\mathrm{-1}}$ ⓒ ${\left(\mathrm{-8}p\right)}^{\mathrm{-1}}.$
ⓐ $\frac{8}{p}$ ⓑ $\frac{1}{8p}$ ⓒ $-\frac{1}{8p}$
Simplify: ⓐ ${11q}^{\mathrm{-1}}$ ⓑ ${\left(11q\right)}^{\mathrm{-1}}$ $\text{\u2212}{\left(11q\right)}^{\mathrm{-1}}$ ⓒ ${\left(\mathrm{-11}q\right)}^{\mathrm{-1}}.$
ⓐ $\frac{1}{11q}$ ⓑ $\frac{1}{11q}$ $-\frac{1}{11q}$ ⓒ $-\frac{1}{11q}$
With negative exponents, the Quotient Rule needs only one form $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$ , for $a\ne 0$ . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.
All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
If $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are real numbers, and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are integers, then
Simplify: ⓐ ${x}^{\mathrm{-4}}\xb7{x}^{6}$ ⓑ ${y}^{\mathrm{-6}}\xb7{y}^{4}$ ⓒ ${z}^{\mathrm{-5}}\xb7{z}^{\mathrm{-3}}.$
Simplify: ⓐ ${x}^{\mathrm{-3}}\xb7{x}^{7}$ ⓑ ${y}^{\mathrm{-7}}\xb7{y}^{2}$ ⓒ ${z}^{\mathrm{-4}}\xb7{z}^{\mathrm{-5}}.$
ⓐ ${x}^{4}$ ⓑ $\frac{1}{{y}^{5}}$ ⓒ $\frac{1}{{z}^{9}}$
Simplify: ⓐ ${a}^{\mathrm{-1}}\xb7{a}^{6}$ ⓑ ${b}^{\mathrm{-8}}\xb7{b}^{4}$ ⓒ ${c}^{\mathrm{-8}}\xb7{c}^{\mathrm{-7}}.$
ⓐ ${a}^{5}$ ⓑ $\frac{1}{{b}^{4}}$ ⓒ $\frac{1}{{c}^{15}}$
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.
Simplify: $\left({m}^{4}{n}^{\mathrm{-3}}\right)\left({m}^{\mathrm{-5}}{n}^{\mathrm{-2}}\right).$
$\begin{array}{cccc}& & & \left({m}^{4}{n}^{\mathrm{-3}}\right)\left({m}^{\mathrm{-5}}{n}^{\mathrm{-2}}\right)\hfill \\ \text{Use the Commutative Property to get like bases together.}\hfill & & & {m}^{4}{m}^{\mathrm{-5}}\xb7{n}^{\mathrm{-2}}{n}^{\mathrm{-3}}\hfill \\ \text{Add the exponents for each base.}\hfill & & & {m}^{\mathrm{-1}}\xb7{n}^{\mathrm{-5}}\hfill \\ \text{Take reciprocals and change the signs of the exponents.}\hfill & & & \frac{1}{{m}^{1}}\xb7\frac{1}{{n}^{5}}\hfill \\ \text{Simplify.}\hfill & & & \frac{1}{m{n}^{5}}\hfill \end{array}$
Simplify: $\left({p}^{6}{q}^{\mathrm{-2}}\right)\left({p}^{\mathrm{-9}}{q}^{\mathrm{-1}}\right).$
$\frac{1}{{p}^{3}{q}^{3}}$
Simplify: $\left({r}^{5}{s}^{\mathrm{-3}}\right)\left({r}^{\mathrm{-7}}{s}^{\mathrm{-5}}\right).$
$\frac{1}{{r}^{2}{s}^{8}}$
If the monomials have numerical coefficients, we multiply the coefficients, just like we did earlier.
Simplify: $\left(2{x}^{\mathrm{-6}}{y}^{8}\right)\left(\mathrm{-5}{x}^{5}{y}^{\mathrm{-3}}\right).$
$\begin{array}{cccc}& & & \left(2{x}^{\mathrm{-6}}{y}^{8}\right)\left(\mathrm{-5}{x}^{5}{y}^{\mathrm{-3}}\right)\hfill \\ \text{Rewrite with the like bases together.}\hfill & & & 2\left(\mathrm{-5}\right)\xb7\left({x}^{\mathrm{-6}}{x}^{5}\right)\xb7\left({y}^{8}{y}^{\mathrm{-3}}\right)\hfill \\ \text{Multiply the coefficients and add the exponents of each variable.}\hfill & & & \mathrm{-10}\xb7{x}^{\mathrm{-1}}\xb7{y}^{5}\hfill \\ \text{Use the definition of a negative exponent,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{\u2212}n}=\frac{1}{{a}^{n}}.\hfill & & & \mathrm{-10}\xb7\frac{1}{{x}^{1}}\xb7{y}^{5}\hfill \\ \text{Simplify.}\hfill & & & \frac{\mathrm{-10}{y}^{5}}{x}\hfill \end{array}$
Simplify: $\left(3{u}^{\mathrm{-5}}{v}^{7}\right)\left(\mathrm{-4}{u}^{4}{v}^{\mathrm{-2}}\right).$
$-\frac{12{v}^{5}}{u}$
Simplify: $\left(\mathrm{-6}{c}^{\mathrm{-6}}{d}^{4}\right)\left(\mathrm{-5}{c}^{\mathrm{-2}}{d}^{\mathrm{-1}}\right).$
$\frac{30{d}^{3}}{{c}^{8}}$
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
Simplify: ${\left(6{k}^{3}\right)}^{\mathrm{-2}}.$
$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(6{k}^{3}\right)}^{\mathrm{-2}}\hfill \\ \text{Use the Product to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(ab\right)}^{m}={a}^{m}{b}^{m}.\hfill & & & \phantom{\rule{4em}{0ex}}{\left(6\right)}^{\mathrm{-2}}{\left({k}^{3}\right)}^{\mathrm{-2}}\hfill \\ \text{Use the Power Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m\xb7n}.\hfill & & & \phantom{\rule{4em}{0ex}}{6}^{\mathrm{-2}}{k}^{\mathrm{-6}}\hfill \\ \text{Use the Definition of a Negative Exponent,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{\u2212}n}=\frac{1}{{a}^{n}}.\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{{6}^{2}}\xb7\frac{1}{{k}^{6}}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{36{k}^{6}}\hfill \end{array}$
Simplify: ${\left(\mathrm{-4}{x}^{4}\right)}^{\mathrm{-2}}.$
$\frac{1}{16{x}^{8}}$
Simplify: ${\left(2{b}^{3}\right)}^{\mathrm{-4}}.$
$\frac{1}{16{b}^{12}}$
Simplify: ${\left(5{x}^{\mathrm{-3}}\right)}^{2}.$
$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(5{x}^{\mathrm{-3}}\right)}^{2}\hfill \\ \\ \\ \text{Use the Product to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(ab\right)}^{m}={a}^{m}{b}^{m}.\hfill & & & \phantom{\rule{4em}{0ex}}{5}^{2}{\left({x}^{\mathrm{-3}}\right)}^{2}\hfill \\ \\ \\ \begin{array}{c}\text{Simplify}\phantom{\rule{0.2em}{0ex}}{5}^{2}\phantom{\rule{0.2em}{0ex}}\text{and multiply the exponents of}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{using the Power}\hfill \\ \text{Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m\xb7n}.\hfill \end{array}\hfill & & & \phantom{\rule{4em}{0ex}}25\xb7{x}^{\mathrm{-6}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite}\phantom{\rule{0.2em}{0ex}}{x}^{\mathrm{-6}}\phantom{\rule{0.2em}{0ex}}\text{by using the Definition of a Negative Exponent,}\hfill \\ {a}^{\text{\u2212}n}=\frac{1}{{a}^{n}}.\hfill \end{array}\hfill & & & \phantom{\rule{4em}{0ex}}25\xb7\frac{1}{{x}^{6}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{25}{{x}^{6}}\hfill \end{array}$
Simplify: ${\left(8{a}^{\mathrm{-4}}\right)}^{2}.$
$\frac{64}{{a}^{8}}$
Simplify: ${\left(2{c}^{\mathrm{-4}}\right)}^{3}.$
$\frac{8}{{c}^{12}}$
To simplify a fraction, we use the Quotient Property and subtract the exponents.
Simplify: $\frac{{r}^{5}}{{r}^{\mathrm{-4}}}.$
$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\frac{{r}^{5}}{{r}^{\mathrm{-4}}}\hfill \\ \text{Use the Quotient Property,}\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\hfill & & & \phantom{\rule{4em}{0ex}}{r}^{5-\left(\mathrm{-4}\right)}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}{r}^{9}\hfill \end{array}$
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