audio read-through 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:

$$ \begin{align} \cssId{s8}{(x^2)^3}\ &\cssId{s9}{=\ (x^2)(x^2)(x^2)}\cr\cr &\cssId{s10}{=\ \overset{\text{three piles, two in each}}{\overbrace{(x\cdot x)(x\cdot x)(x\cdot x)}}}\cr\cr &\cssId{s11}{=\ x^6} \cssId{s12}{= x^{2\cdot 3}} \end{align} $$

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.


$x^mx^n = x^{m+n}$ 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


$(x^3)^2 = x^p\,$ where $\,p = \text{?}$
Answer: $p = 6$