Practice With
$\,\displaystyle x^{-p} = \frac{1}{x^p}$
Recall that $\displaystyle\,x^{-1} = \frac 1x\,.$ That is, $\displaystyle\,x^{-1}\,$ is the reciprocal of $\,x\,.$
It follows, using the exponent laws, that:
$$\cssId{s8}{x^{-p} = (x^p)^{-1} = \frac{1}{x^p}}$$That is, $\,x^{-p}\,$ is the reciprocal of $\,x^p\,.$
Continuing, it's convenient to notice that expressions of the form $\,x^m\,$ can be moved from numerator to denominator, or from denominator to numerator, just by changing the sign of the exponent.
For example:
$\displaystyle \cssId{s17}{\frac{1}{x^{-3}}} \cssId{s18}{= \frac{x^3}{1}} \cssId{s19}{= x^3}\,$: exponent was negative in denominator; is positive in numerator
$\displaystyle \cssId{s22}{\frac{1}{x^{3}}} \cssId{s23}{= \frac{x^{-3}}{1}} \cssId{s24}{= x^{-3}}\,$: exponent was positive in denominator; is negative in numerator
$\displaystyle \cssId{s27}{x^3} \cssId{s28}{= \frac{x^3}{1}} \cssId{s29}{= \frac{1}{x^{-3}}}\,$: exponent was positive in numerator; is negative in denominator
$\displaystyle \cssId{s32}{x^{-3}} \cssId{s33}{= \frac{x^{-3}}{1}} \cssId{s34}{= \frac{1}{x^3}}\,$: exponent was negative in numerator; is positive in denominator
All the exponent laws are stated below, for completeness.
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 |
| $\displaystyle \frac{x^m}{x^n} = x^{m-n}$ | 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 |