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$\begin{array}{ll}\frac{1}{{x}^{-3}{y}^{-2}{z}^{-1}}.\hfill & \text{This}\text{\hspace{0.17em}}\text{fraction}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{written}\text{\hspace{0.17em}}\text{without}\text{\hspace{0.17em}}\text{negative}\text{\hspace{0.17em}}\text{exponents}\hfill \\ \hfill & \text{by}\text{\hspace{0.17em}}\text{moving}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}factors\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{denominator}\text{\hspace{0.17em}}\text{to}\hfill \\ \hfill & \text{the}\text{\hspace{0.17em}}\text{numerator}\text{.}\text{\hspace{0.17em}}\text{Change}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{each}\text{\hspace{0.17em}}\text{exponent:}\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}+3,\hfill \\ \hfill & -2\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}+2,\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}+1.\hfill \\ \frac{1}{{x}^{-3}{y}^{-2}{z}^{-1}}={x}^{3}{y}^{2}{z}^{1}={x}^{3}{y}^{2}z\hfill & \hfill \end{array}$
Write each of the following so that only positive exponents appear.
$\frac{6{m}^{-3}{n}^{-2}}{7{k}^{-1}}$
$\frac{6k}{7{m}^{3}{n}^{2}}$
$\frac{1}{{a}^{-2}{b}^{-6}{c}^{-8}}$
${a}^{2}{b}^{6}{c}^{8}$
$\frac{3a{(a-5b)}^{-2}}{5b{(a-4b)}^{5}}$
$\frac{3a}{5b{(a-5b)}^{2}{(a-4b)}^{5}}$
Rewrite $\frac{24{a}^{7}{b}^{9}}{{2}^{3}{a}^{4}{b}^{-6}}$ in a simpler form.
Notice that we are dividing powers with the same base. We’ll proceed by using the rules of exponents.
$\begin{array}{lllll}\frac{24{a}^{7}{b}^{9}}{{2}^{3}{a}^{4}{b}^{-6}}\hfill & =\hfill & \frac{24{a}^{7}{b}^{9}}{8{a}^{4}{b}^{-6}}\hfill & =\hfill & 3{a}^{7-4}{b}^{9-(-6)}\hfill \\ \hfill & \hfill & \hfill & =\hfill & 3{a}^{3}{b}^{9+6}\hfill \\ \hfill & \hfill & \hfill & =\hfill & 3{a}^{3}{b}^{15}\hfill \end{array}$
Write $\frac{9{a}^{5}{b}^{3}}{5{x}^{3}{y}^{2}}$ so that no denominator appears.
We can eliminate the denominator by moving all factors that make up the denominator to the numerator.
$9{a}^{5}{b}^{3}{5}^{-1}{x}^{-3}{y}^{-2}$
Find the value of $\frac{1}{{10}^{-2}}+\frac{3}{{4}^{-3}}$ .
We can evaluate this expression by eliminating the negative exponents.
$\begin{array}{lll}\frac{1}{{10}^{-2}}+\frac{3}{{4}^{-3}}\hfill & =\hfill & 1\cdot {10}^{2}+3\cdot {4}^{3}\hfill \\ \hfill & =\hfill & 1\cdot 100+3\cdot 64\hfill \\ \hfill & =\hfill & 100+192\hfill \\ \hfill & =\hfill & 292\hfill \end{array}$
Rewrite $\frac{36{x}^{8}{b}^{3}}{{3}^{2}{x}^{-2}{b}^{-5}}$ in a simpler form.
$4{x}^{10}{b}^{8}$
Write $\frac{{2}^{4}{m}^{-3}{n}^{7}}{{4}^{-1}{x}^{5}}$ in a simpler form and one in which no denominator appears.
$64{m}^{-3}{n}^{7}{x}^{-5}$
Find the value of $\frac{2}{{5}^{-2}}+{6}^{-2}\cdot {2}^{3}\cdot {3}^{2}$ .
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Write the following expressions using only positive exponents. Assume all variables are nonzero.
${x}^{-4}$
${a}^{-8}$
${b}^{-12}$
${y}^{-1}$
${(x+1)}^{-2}$
${(y-4)}^{-6}$
${(r+3)}^{-8}$
${x}^{3}{y}^{-2}$
${a}^{4}{b}^{-1}$
${a}^{2}{b}^{3}{c}^{-2}$
${x}^{3}{y}^{-4}{z}^{2}w$
${a}^{3}{b}^{-1}z{w}^{2}$
${x}^{4}{y}^{-8}{z}^{-3}{w}^{-4}$
${a}^{-4}{b}^{-6}{c}^{-1}{d}^{4}$
$\frac{{d}^{4}}{{a}^{4}{b}^{6}c}$
${x}^{9}{y}^{-6}{z}^{-1}{w}^{-5}{r}^{-2}$
$5{x}^{2}{y}^{2}{z}^{-5}$
$4{x}^{3}{(x+1)}^{2}{y}^{-4}{z}^{-1}$
$7{a}^{2}{(a-4)}^{3}{b}^{-6}{c}^{-7}$
$\frac{7{a}^{2}{\left(a-4\right)}^{3}}{{b}^{6}{c}^{7}}$
$18{b}^{-6}{({b}^{2}-3)}^{-5}{c}^{-4}{d}^{5}{e}^{-1}$
$7{(w+2)}^{-2}{(w+1)}^{3}$
$\frac{7{\left(w+1\right)}^{3}}{{\left(w+2\right)}^{2}}$
$2{(a-8)}^{-3}{(a-2)}^{5}$
${({x}^{2}+3)}^{3}{({x}^{2}-1)}^{-4}$
$\frac{{\left({x}^{2}+3\right)}^{3}}{{\left({x}^{2}-1\right)}^{4}}$
${({x}^{4}+2x-1)}^{-6}{(x+5)}^{4}$
${(3{x}^{2}-4x-8)}^{-9}{(2x+11)}^{-3}$
$\frac{1}{{\left(3{x}^{2}-4x-8\right)}^{9}{\left(2x+11\right)}^{3}}$
${(5{y}^{2}+8y-6)}^{-2}{(6y-1)}^{-7}$
$7a{({a}^{2}-4)}^{-2}{({b}^{2}-1)}^{-2}$
$\frac{7a}{{\left({a}^{2}-4\right)}^{2}{\left({b}^{2}-1\right)}^{2}}$
${(x-5)}^{-4}3{b}^{2}{c}^{4}{(x+6)}^{8}$
${({y}^{3}+1)}^{-1}5{y}^{3}{z}^{-4}{w}^{-2}{({y}^{3}-1)}^{-2}$
$\frac{5{y}^{3}}{\left({y}^{3}+1\right){z}^{4}{w}^{2}{\left({y}^{3}-1\right)}^{2}}$
$5{x}^{3}(2{x}^{-7})$
$6{a}^{-4}(2{a}^{-6})$
${5}^{-1}{a}^{-2}{b}^{-6}{b}^{-11}{c}^{-3}{c}^{9}$
$7{a}^{-3}{b}^{-9}\cdot 5a{}^{6}b{c}^{-2}{c}^{4}$
${(x+5)}^{2}{(x+5)}^{-6}$
$\frac{1}{{\left(x+5\right)}^{4}}$
${(a-4)}^{3}{(a-4)}^{-10}$
$8{(b+2)}^{-8}{(b+2)}^{-4}{(b+2)}^{3}$
$\frac{8}{{\left(b+2\right)}^{9}}$
$3{a}^{5}{b}^{-7}{({a}^{2}+4)}^{-3}6{a}^{-4}b{({a}^{2}+4)}^{-1}({a}^{2}+4)$
$-4{a}^{3}{b}^{-5}(2{a}^{2}{b}^{7}{c}^{-2})$
$\frac{-8{a}^{5}{b}^{2}}{{c}^{2}}$
$-2{x}^{-2}{y}^{-4}{z}^{4}(-6{x}^{3}{y}^{-3}z)$
${(-9)}^{-3}{(9)}^{3}$
${(4)}^{2}{(2)}^{-4}$
$\frac{1}{{a}^{-1}}$
$\frac{7}{{x}^{-8}}$
$\frac{6}{{a}^{2}{b}^{-4}}$
$\frac{3{c}^{5}}{{a}^{3}{b}^{-3}}$
$\frac{3{b}^{3}{c}^{5}}{{a}^{3}}$
$\frac{16{a}^{-2}{b}^{-6}c}{2y{z}^{-5}{w}^{-4}}$
$\frac{24{y}^{2}{z}^{-8}}{6{a}^{2}{b}^{-1}{c}^{-9}{d}^{3}}$
$\frac{4b{c}^{9}{y}^{2}}{{a}^{2}{d}^{3}{z}^{8}}$
$\frac{{3}^{-1}{b}^{5}{(b+7)}^{-4}}{{9}^{-1}{a}^{-4}{(a+7)}^{2}}$
$\frac{36{a}^{6}{b}^{5}{c}^{8}}{{3}^{2}{a}^{3}{b}^{7}{c}^{9}}$
$\frac{4{a}^{3}}{{b}^{2}c}$
$\frac{45{a}^{4}{b}^{2}{c}^{6}}{15{a}^{2}{b}^{7}{c}^{8}}$
$\frac{{3}^{3}{x}^{4}{y}^{3}z}{{3}^{2}x{y}^{5}{z}^{5}}$
$\frac{3{x}^{3}}{{y}^{2}{z}^{4}}$
$\frac{21{x}^{2}{y}^{2}{z}^{5}{w}^{4}}{7xy{z}^{12}{w}^{14}}$
$\frac{33{a}^{-4}{b}^{-7}}{11{a}^{3}{b}^{-2}}$
$\frac{3}{{a}^{7}{b}^{5}}$
$\frac{51{x}^{-5}{y}^{-3}}{3xy}$
$\frac{{2}^{6}{x}^{-5}{y}^{-2}{a}^{-7}{b}^{5}}{{2}^{-1}{x}^{-4}{y}^{-2}{b}^{6}}$
$\frac{128}{{a}^{7}bx}$
$\frac{{(x+3)}^{3}{(y-6)}^{4}}{{(x+3)}^{5}{(y-6)}^{-8}}$
$\frac{5{x}^{4}{y}^{3}}{{a}^{3}}$
$\frac{23{a}^{4}{b}^{5}{c}^{-2}}{{x}^{-6}{y}^{5}}$
$\frac{23{a}^{4}{b}^{5}{x}^{6}}{{c}^{2}{y}^{5}}$
$\frac{{2}^{3}{b}^{5}{c}^{2}{d}^{-9}}{4{b}^{4}cx}$
$\frac{10{x}^{3}{y}^{-7}}{3{x}^{5}{z}^{2}}$
$\frac{10}{3{x}^{2}{y}^{7}{z}^{2}}$
$\frac{3{x}^{2}{y}^{-2}(x-5)}{{9}^{-1}{(x+5)}^{3}}$
$\frac{14{a}^{2}{b}^{2}{c}^{-12}{({a}^{2}+21)}^{-4}}{{4}^{-2}{a}^{2}{b}^{-1}{(a+6)}^{3}}$
$\frac{224{b}^{3}}{{c}^{12}{\left({a}^{2}+21\right)}^{4}{\left(a+6\right)}^{3}}$
For the following problems, evaluate each numerical expression.
${4}^{-1}$
${6}^{-2}$
${3}^{-4}$
$4\cdot {9}^{-2}$
${2}^{-3}({3}^{-2})$
${10}^{-2}+3({10}^{-2})$
${(-10)}^{-1}$
$\frac{{4}^{-1}}{{5}^{-2}}$
$\frac{{2}^{-1}+{4}^{-1}}{{2}^{-2}+{4}^{-2}}$
For the following problems, write each expression so that only positive exponents appear.
${({a}^{6})}^{-2}$
${({x}^{7})}^{-4}$
${({b}^{-2})}^{7}$
${({y}^{-3})}^{-4}$
${({a}^{-1})}^{-1}$
$\begin{array}{ll}{({a}^{0})}^{-1},\hfill & a\ne 0\hfill \end{array}$
$\begin{array}{ll}{({m}^{0})}^{-1},\hfill & m\ne 0\hfill \end{array}$
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${({x}^{-3}{y}^{7})}^{-4}$
${({x}^{6}{y}^{6}{z}^{-1})}^{2}$
$\frac{{x}^{12}{y}^{12}}{{z}^{2}}$
${({a}^{-5}{b}^{-1}{c}^{0})}^{6}$
${\left(\frac{{y}^{3}}{{x}^{-4}}\right)}^{5}$
${x}^{20}{y}^{15}$
${\left(\frac{{a}^{-8}}{{b}^{-6}}\right)}^{3}$
${\left(\frac{2a}{{b}^{3}}\right)}^{4}$
$\frac{16{a}^{4}}{{b}^{12}}$
${\left(\frac{3b}{{a}^{2}}\right)}^{-5}$
${\left(\frac{{5}^{-1}{a}^{3}{b}^{-6}}{{x}^{-2}{y}^{9}}\right)}^{2}$
$\frac{{a}^{6}{x}^{4}}{25{b}^{12}{y}^{18}}$
${\left(\frac{4{m}^{-3}{n}^{6}}{2{m}^{-5}n}\right)}^{3}$
${\left(\frac{{r}^{5}{s}^{-4}}{{m}^{-8}{n}^{7}}\right)}^{-4}$
$\frac{{n}^{28}{s}^{16}}{{m}^{32}{r}^{20}}$
${\left(\frac{{h}^{-2}{j}^{-6}}{{k}^{-4}p}\right)}^{-5}$
( [link] ) Simplify ${(4{x}^{5}{y}^{3}{z}^{0})}^{3}$
$64{x}^{15}{y}^{9}$
( [link] ) Find the sum. $-15+3$ .
( [link] ) Simplify $(-3)(-8)+4(-5)$ .
( [link] ) Find the value of $m$ if $m=\frac{-3k-5t}{kt+6}$ when $k=4$ and $t=-2$ .
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