BASIC EXPONENT PRACTICE WITH FRACTIONS
• PRACTICE (online exercises and printable worksheets)
 DEFINITIONS:   properties of exponents base; exponent; power Let $\,x\in\Bbb{R}\,$. In the expression $\,x^n\,$, $\,x\,$ is called the base and $\,n\,$ is called the exponent or the power. positive integers If $\,n\in\{1,2,3,\ldots\}\,$, then $\,x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,$, where there are $\,n\,$ factors in the product. In this case, $\,x^n\,$ is just a shorthand for repeated multiplication. Note that $\,x^1 = x\,$ for all real numbers $\,x\,$. zero If $\,x\ne 0\,$, then $\,x^0 = 1\,$. The expression $\,0^0\,$ is not defined. negative integers If $\,n\in\{1,2,3,\ldots\}\,$ and $\,x\ne 0\,$, then $\displaystyle\,x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}$, where there are $\,n\,$ factors in the product. In particular, $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,$. That is, $\,x^{-1}\,$ is the reciprocal of $\,x\,$.

With fractions, it looks like this:

$\displaystyle (\frac{a}{b})^{-1} = \frac{1}{\frac{a}{b}} = 1 \div \frac{a}{b} = 1\cdot\frac{b}{a} = \frac{b}{a}$

That is, the reciprocal of $\displaystyle\,\frac{a}{b}\,$ is $\displaystyle\,\frac{b}{a}\,$.

Now that you've mired through this calculation once,
you'll never have to do it this long way again!

When a fraction is raised to the $\,-1\,$ power, it just flips.
The numerator becomes the new denominator, and the denominator becomes the new numerator.

EXAMPLES:
$\displaystyle \left(\frac23\right)^{-1} = \frac32$
$\displaystyle \left(\frac23\right)^{0} = 1$
$\displaystyle \left(\frac23\right)^{1} = \frac23$
$\displaystyle \left(-\frac23\right)^{-1} = -\frac32$
$\displaystyle 5^{-1} = \frac15$
$\displaystyle (-5)^{-1} = -\frac15$
$\displaystyle \left(\frac13\right)^{-1} = 3$
$\displaystyle \left(-\frac13\right)^{-1} = -3$
Master the ideas from this section
Practice with$\,x^mx^n = x^{m+n}$