Recall that
$\displaystyle\,x^{-1} = \frac 1x\,$.
That is,
$\displaystyle\,x^{-1}\,$ is the reciprocal of $\,x\,$.
It follows, using the exponent laws,
that
$\displaystyle x^{-p} = (x^p)^{-1} = \frac{1}{x^p}\,$.
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.
$x^mx^n = x^{m+n}$ | 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 |