audio read-through Practice With $\,\displaystyle\frac{x^m}{x^n} = x^{m-n}$

All the exponent laws are stated below, for completeness.

This web exercise gives practice with:

$$ \cssId{s6}{\frac{x^m}{x^n} = x^{m-n}} $$

Here's the motivation for this exponent law:

$$ \begin{align} \cssId{s8}{\frac{x^5}{x^2}}\ &\cssId{s9}{=\ \frac{x\cdot x\cdot x\cdot \cancel{x}\cdot \bcancel{x}}{\cancel{x}\cdot \bcancel{x}}}\cr\cr &\cssId{s10}{=\ x\cdot x\cdot x}\cr\cr &\cssId{s11}{=\ x^3} \cssId{s12}{= x^{5-2}} \end{align} $$

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
$\displaystyle\large \frac{x^m}{x^n} = x^{m-n}$ 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


$\displaystyle \frac{x^5}{x^3} = x^p\,$ where $\,p = \text{?}$
Answer: $p = 2$