Recall that
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$\displaystyle\,x^{-1} = \frac 1x\,$. That is,
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$\displaystyle\,x^{-1}\,$ is the reciprocal of $\,x\,$.
It follows, using the exponent laws, that
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$\displaystyle x^{-p} = (x^p)^{-1} = \frac{1}{x^p}\,$.
That is,
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$\,x^{-p}\,$ is the reciprocal of $\,x^p\,$.
Continuing, it's convenient to notice that expressions of the form
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$\,x^m\,$
can be moved from numerator to denominator, or from denominator to numerator,
just by changing the sign of the exponent.
For example:
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$\displaystyle
\frac{1}{x^{-3}} = \frac{x^3}{1} = x^3\,$:
exponent was negative in denominator; is positive in numerator
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$\displaystyle
\frac{1}{x^{3}} = \frac{x^{-3}}{1} = x^{-3}\,$:
exponent was positive in denominator; is negative in numerator
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$\displaystyle
x^3 = \frac{x^3}{1} = \frac{1}{x^{-3}}\,$:
exponent was positive in numerator; is negative in denominator
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$\displaystyle
x^{-3} = \frac{x^{-3}}{1} = \frac{1}{x^3}\,$:
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 |