2.6 The power rules for exponents

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

Using the product rule we get

$\begin{array}{l}{(a^2)}^3=a^{2+2+2}\hfill \\ {(a^2)}^3=a^{3\cdot 2}\hfill \\ {(a^2)}^3=a^6\hfill \end{array}$

$\begin{array}{lll}{(x^9)}^4\hfill & =\hfill & x^9\cdot x^9\cdot x^9\cdot x^9\hfill \\ {(x^9)}^4\hfill & =\hfill & x^{9+9+9+9}\hfill \\ {(x^9)}^4\hfill & =\hfill & x^{4\cdot 9}\hfill \\ {(x^9)}^4\hfill & =\hfill & x^{36}\hfill \end{array}$

$\begin{array}{cc}{(x^3)}^4=\begin{array}{||}\hline x^{3\cdot 4}\\ \hline\end{array}{x}^{12}& \text{The}\text{box}\text{represents}\text{a}\text{step}\text{done}\text{mentally.}\end{array}$

$${(y^5)}^3=\begin{array}{||}\hline y^{5\cdot 3}\\ \hline\end{array}=y^{15}$$

$${(d^{20})}^6=\begin{array}{||}\hline d^{20\cdot 6}\\ \hline\end{array}=d^{120}$$

${(x^{\square })}^{△}=x^{\square △}$

$\begin{array}{llll}{(ab)}^3\hfill & =\hfill & ab\cdot ab\cdot ab\hfill & \text{Use}\text{}\text{the}\text{commutative}\text{property}\text{of}\text{multiplication}.\hfill \\ \hfill & =\hfill & aaabbb\hfill & \hfill \\ \hfill & =\hfill & a^3{b}^3\hfill & \hfill \end{array}$

$\begin{array}{lll}{(xy)}^5\hfill & =\hfill & xy\cdot xy\cdot xy\cdot xy\cdot xy\hfill \\ \hfill & =\hfill & xxxxx\cdot yyyyy\hfill \\ \hfill & =\hfill & x^5{y}^5\hfill \end{array}$

$\begin{array}{lll}{(4xy\mathrm{z})}^2\hfill & =\hfill & 4xyz\cdot 4xyz\hfill \\ \hfill & =\hfill & 4\cdot 4\cdot xx\cdot yy\cdot zz\hfill \\ \hfill & =\hfill & 16x^2{y}^2{z}^2\hfill \end{array}$

${(ab)}^7=a^7{b}^7$

${(axy)}^4=a^4{x}^4{y}^4$

$\begin{array}{cc}{(3ab)}^2=3^2{a}^2{b}^2=9a^2{b}^2& \text{Don't}\text{forget}\text{to}\text{apply}\text{the}\text{exponent}\text{to}\text{the}\text{3!}\end{array}$

$\begin{array}{c}{(2st)}^5=2^5{s}^5{t}^5=32s^5{t}^5\end{array}$

$\begin{array}{ll}{(a{b}^3)}^2=a^2{(b^3)}^2=a^2{b}^6\hfill & \text{We}\text{used}\text{two}\text{rules}\text{here}\text{.}\text{First,}\text{the}\text{power}\text{rule}\text{for}\hfill \\ \hfill & \text{products}\text{.}\text{Second,}\text{the}\text{power}\text{rule}\text{for}\text{powers.}\hfill \end{array}$

$\begin{array}{ll}{(7a^4{b}^2{c}^8)}^2\hfill & =7^2{(a^4)}^2{(b^2)}^2{(c^8)}^2\hfill \\ \hfill & =49a^8{b}^4{c}^{16}\hfill \end{array}$

$\begin{array}{cc}\text{If}6a^3{c}^7\ne 0,\text{then}{(6a^3{c}^7)}^0=1& \text{Recall}\text{that}{x}^0=1\text{for}x\ne 0.\end{array}$

$\begin{array}{ll}{\left[2{\left(x+1\right)}^4\right]}^6\hfill & =2^6{(x+1)}^{24}\hfill \\ \hfill & =64{(x+1)}^{24}\hfill \end{array}$

${\left(\frac{a}{b}\right)}^3=\frac{a}{b}\cdot \frac{a}{b}\cdot \frac{a}{b}=\frac{a\cdot a\cdot a}{b\cdot b\cdot b}=\frac{a^3}{b^3}$

${\left(\frac{x}{y}\right)}^6=\frac{x^6}{y^6}$

${\left(\frac{a}{c}\right)}^2=\frac{a^2}{c^2}$

${\left(\frac{2x}{b}\right)}^4=\frac{{\left(2x\right)}^4}{b^4}=\frac{2^4{x}^4}{b^4}=\frac{16x^4}{b^4}$

${\left(\frac{a^3}{b^5}\right)}^7=\frac{{(a^3)}^7}{{(b^5)}^7}=\frac{a^{21}}{b^{35}}$

$\begin{array}{ccc}{\left(\frac{3c^4{r}^2}{2^3{g}^5}\right)}^3=\frac{3^3{c}^{12}{r}^6}{2^9{g}^{15}}=\frac{27c^{12}{r}^6}{2^9{g}^{15}}& \text{or}& \frac{27c^{12}{r}^6}{512g^{15}}\end{array}$

${\left[\frac{\left(a-2\right)}{\left(a+7\right)}\right]}^4=\frac{{\left(a-2\right)}^4}{{\left(a+7\right)}^4}$

${\left[\frac{6x{\left(4-x\right)}^4}{2a{\left(y-4\right)}^6}\right]}^2=\frac{6^2{x}^2{\left(4-x\right)}^8}{2^2{a}^2{\left(y-4\right)}^{12}}=\frac{36x^2{\left(4-x\right)}^8}{4a^2{\left(y-4\right)}^{12}}=\frac{9x^2{\left(4-x\right)}^8}{a^2{\left(y-4\right)}^{12}}$

$\begin{array}{llll}{\left(\frac{a^3{b}^5}{a^{2}b}\right)}^3\hfill & =\hfill & {(a^{3-2}{b}^{5-1})}^3\hfill & \text{We}\text{can}\text{simplify}\text{within}\text{the}\text{parentheses}\text{.}\text{We}\hfill \\ \hfill & \hfill & \hfill & \text{have}\text{a}\text{rule}\text{that}\text{tells}\text{us}\text{to}\text{proceed}\text{this}\text{way}\text{.}\hfill \\ \hfill & =\hfill & {(a{b}^4)}^3\hfill & \hfill \\ \hfill & =\hfill & a^3{b}^{12}\hfill & \hfill \\ {\left(\frac{a^3{b}^5}{a^{2}b}\right)}^3\hfill & =\hfill & \frac{a^9{b}^{15}}{a^6{b}^3}=a^{9-6}{b}^{15-3}=a^3{b}^{12}\hfill & \text{We}\text{could}\text{have}\text{actually}\text{used}\text{the}\text{power}\text{rule}\text{for}\hfill \\ \hfill & \hfill & \hfill & \text{quotients}\text{first}\text{.}\text{Distribute}\text{the}\text{exponent,}\text{then}\hfill \\ \hfill & \hfill & \hfill & \text{simplify}\text{using}\text{the}\text{other}\text{rules}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{It}\text{is}\text{probably}\text{better,}\text{for}\text{the}\text{sake}\text{of}\text{consistency,}\hfill \\ \hfill & \hfill & \hfill & \text{to}\text{work}\text{inside}\text{the}\text{parentheses}\text{first.}\hfill \end{array}$

${\left(\frac{a^r{b}^s}{c^t}\right)}^w=\frac{a^{rw}{b}^{sw}}{c^{tw}}$