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:
$$ \begin{align} \cssId{s8}{\frac{x^5}{x^2}}\ &\cssId{s9}{=\ \frac{x\cdot x\cdot x\cdot \cancel{x}\cdot \bcancel{x}}{\cancel{x}\cdot \bcancel{x}}}\cr\cr &\cssId{s10}{=\ x\cdot x\cdot x}\cr\cr &\cssId{s11}{=\ x^3} \cssId{s12}{= x^{5-2}} \end{align} $$
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 |
$\displaystyle\large \frac{x^m}{x^n} = x^{m-n}$ | 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 |
Example
$\displaystyle
\frac{x^5}{x^3} = x^p\,$ where $\,p = \text{?}$
Answer:
$p = 2$