Practice With $\,x^mx^n = x^{m+n}$
All the exponent laws are stated below, for completeness.
This web exercise gives practice with:
$$ \cssId{s6}{x^mx^n = x^{m+n}} $$Here's the motivation for this exponent law:
$$ \begin{align} \cssId{s8}{x^2} \cssId{s9}{x^3}\ &\cssId{s10}{= \overset{\text{two factors}}{\overbrace{x\cdot x}}} \cssId{s11}{\cdot \overset{\text{three factors}}{\overbrace{x\cdot x\cdot x}}}\cr\cr &\cssId{s12}{\ = \ \overset{\text{five factors}}{\overbrace{x\cdot x\cdot x\cdot x\cdot x}}}\cr\cr &\cssId{s13}{\ = \ x^5} \cssId{s14}{\ = \ x^{2+3}} \end{align} $$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.
Then:
| $\large x^mx^n = x^{m+n}$ | 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 |
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Example
