﻿ Basic Exponent Practice with Fractions
BASIC EXPONENT PRACTICE WITH FRACTIONS
by Dr. Carol JVF Burns (website creator)
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• PRACTICE (online exercises and printable worksheets)
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 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 $\displaystyle\, \cssId{s26}{x^{-n}} \cssId{s27}{= \frac{1}{x^n}} \cssId{s28}{= \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}}$, where there are $\,n\,$ factors in the product. In particular, $\,\displaystyle \cssId{s31}{x^{-1} = \frac{1}{x}}\,$ for all nonzero real numbers $\,x\,$. That is, $\,x^{-1}\,$ is the reciprocal of $\,x\,$.

With fractions, it looks like this:

$\displaystyle \cssId{s36}{(\frac{a}{b})^{-1}} \cssId{s37}{= \frac{1}{\frac{a}{b}}} \cssId{s38}{= 1 \div \frac{a}{b}} \cssId{s39}{= 1\cdot\frac{b}{a}} \cssId{s40}{= \frac{b}{a}}$

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 $\,x^mx^n = x^{m+n}$