Practice With $\,(x^m)^n = x^{mn}$
All the exponent laws are stated below, for completeness.
This web exercise gives practice with:
$$ \cssId{s6}{(x^m)^n = x^{mn}} $$Here's the motivation for this exponent law:
$$ \begin{align} \cssId{s8}{(x^2)^3}\ &\cssId{s9}{=\ (x^2)(x^2)(x^2)}\cr\cr &\cssId{s10}{=\ \overset{\text{three piles, two in each}}{\overbrace{(x\cdot x)(x\cdot x)(x\cdot x)}}}\cr\cr &\cssId{s11}{=\ x^6} \cssId{s12}{= x^{2\cdot 3}} \end{align} $$
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 |
$\large (x^m)^n = x^{mn}$ | 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 |
Example
$(x^3)^2 = x^p\,$ where $\,p = \text{?}$
Answer:
$p = 6$