# Multi-step Exponent Law Practice

Need some basic practice with exponent laws first?

In this exercise you will practice with the exponent laws, all mixed-up.

These problems require the application of more than one exponent law. For simpler problems,

EXPONENT LAWS

Let $\,x\,,$ $\,y\,,$ $\,m\,,$ and $\,n\,$ be real numbers, with the following exceptions:

• a base and exponent cannot simultaneously be zero (since $\,0^0\,$ is undefined);
• division by zero is not allowed;
• for non-integer exponents (like $\,\frac12\,$ or $\,0.4\,$), assume that bases are positive.

Then:

 $x^mx^n = x^{m+n}$ Verbalize: same base, things multiplied, add the exponents $\displaystyle \frac{x^m}{x^n} = x^{m-n}$ Verbalize: same base, things divided, subtract the exponents $(x^m)^n = x^{mn}$ Verbalize: something to a power, to a power; multiply the exponents $(xy)^m = x^my^m$ Verbalize: product to a power; each factor gets raised to the power $\displaystyle \left(\frac{x}{y}\right)^m = \frac{x^m}{y^m}$ Verbalize: fraction to a power; both numerator and denominator get raised to the power

## Examples

\displaystyle \begin{align} \cssId{s42}{\left(\frac{1}{x^2}\right)^3} &\cssId{s43}{= (x^{-2})^3} \cssId{s44}{= x^{-2\,\cdot\, 3}}\cr &\cssId{s45}{= x^{-6}} \cssId{s46}{= x^p} \end{align}

where $\,p = -6$
\displaystyle \begin{align} \cssId{s48}{\left(\frac{x^2}{x^3}\right)^5} &\cssId{s49}{= (x^{2-3})^5} \cssId{s50}{= (x^{-1})^5}\cr &\cssId{s51}{= x^{-1\,\cdot\, 5}} \cssId{s52}{= x^{-5}} \cssId{s53}{= x^p} \end{align}

where $\,p = -5$
\begin{align} \cssId{s55}{(x^2x^4)^{-1}}\ &\cssId{s56}{= (x^{2+4})^{-1}} \cssId{s57}{= (x^6)^{-1}}\cr\cr &\cssId{s58}{= x^{6\,\cdot\, (-1)}} \cssId{s59}{= x^{-6}} \cssId{s60}{= x^p} \end{align}

where $\,p = -6$
\displaystyle \begin{align} \cssId{s62}{\frac{x^2x^{-3}}{x^5}}\ &\cssId{s63}{= \frac{x^{2 + (-3)}}{x^5}} \cssId{s64}{= \frac{x^{-1}}{x^5}}\cr\cr &\cssId{s65}{= x^{-1-5}} \cssId{s66}{= x^{-6}} \cssId{s67}{= x^p} \end{align}

where $\,p = -6$
\displaystyle \begin{align} \cssId{s69}{\frac{x^2}{x^3x^4}}\ &\cssId{s70}{= \frac{x^2}{x^{3+4}}} \cssId{s71}{= \frac{x^2}{x^7}}\cr\cr &\cssId{s72}{= x^{2-7}} \cssId{s73}{= x^{-5}} \cssId{s74}{= x^p} \end{align}

where $\,p = -5$
\displaystyle \begin{align} \cssId{s76}{\frac{(x^2)^3}{(x^{-1})^4}}\ &\cssId{s77}{= \frac{x^{2\,\cdot\,3}}{x^{-1\,\cdot\,4}}} \cssId{s78}{= \frac{x^6}{x^{-4}}}\cr\cr &\cssId{s79}{= x^{6-(-4)}} \cssId{s80}{= x^{10}} \cssId{s81}{= x^p} \end{align}

where $\,p = 10$