PRACTICE WITH $\,(x^m)^n = x^{mn}$

by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

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

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
$(x^m)^n = x^{mn}$ 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: $p = 6$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Practice with $\,x^m/x^n = x^{m-n}$

(an even number, please)