BASIC EXPONENT PRACTICE WITH FRACTIONS
DEFINITIONS:   properties of exponents
base;
exponent;
power
Let [beautiful math coming... please be patient] $\,x\in\Bbb{R}\,$.
In the expression [beautiful math coming... please be patient] $\,x^n\,$,
$\,x\,$ is called the base
and $\,n\,$ is called the exponent or the power.
positive integers If [beautiful math coming... please be patient] $\,n\in\{1,2,3,\ldots\}\,$,
then [beautiful math coming... please be patient] $\,x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,$,
where there are $\,n\,$ factors in the product.

In this case, [beautiful math coming... please be patient] $\,x^n\,$ is just a shorthand for repeated multiplication.

Note that [beautiful math coming... please be patient] $\,x^1 = x\,$ for all real numbers $\,x\,$.
zero If [beautiful math coming... please be patient] $\,x\ne 0\,$, then $\,x^0 = 1\,$.
The expression $\,0^0\,$ is not defined.
negative integers If [beautiful math coming... please be patient] $\,n\in\{1,2,3,\ldots\}\,$ and $\,x\ne 0\,$,
then [beautiful math coming... please be patient] $\displaystyle\,x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}$,
where there are $\,n\,$ factors in the product.
In particular, [beautiful math coming... please be patient] $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,$.
That is, [beautiful math coming... please be patient] $\,x^{-1}\,$ is the reciprocal of $\,x\,$.

With fractions, it looks like this:

[beautiful math coming... please be patient] $\displaystyle (\frac{a}{b})^{-1} = \frac{1}{\frac{a}{b}} = 1 \div \frac{a}{b} = 1\cdot\frac{b}{a} = \frac{b}{a}$

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
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Practice with $\,x^mx^n = x^{m+n}$

 
 

As needed, input your answer as a diagonal fraction (e.g., 2/3), since you can't type horizontal fractions.

Simplify:
    
(an even number, please)