audio read-through Basic Exponent Practice with Fractions

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 $$\cssId{s15}{x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,,}$$ 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 $$ \cssId{s26}{x^{-n}} \cssId{s27}{= \frac{1}{x^n}} \cssId{s28}{= \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}}\,, $$ where there are $\,n\,$ factors in the product.

In particular, $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,$. That is, $\,x^{-1}\,$ is the reciprocal of $\,x\,$.

With fractions, it looks like this:

$$\begin{align} \cssId{s36}{(\frac{a}{b})^{-1}}\ &\ \cssId{s37}{= \frac{1}{\frac{a}{b}}} \cssId{s38}{= 1 \div \frac{a}{b}}\cr\cr &\ \cssId{s39}{= 1\cdot\frac{b}{a}} \cssId{s40}{= \frac{b}{a}} \end{align} $$

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

$\displaystyle \left(\frac23\right)^{-1} = \frac32$
$\displaystyle \left(\frac23\right)^{0} = 1$
$\displaystyle \left(\frac23\right)^{1} = \frac23$
$\displaystyle \left(-\frac23\right)^{-1} = -\frac32$
$\displaystyle 5^{-1} = \frac15$
$\displaystyle (-5)^{-1} = -\frac15$
$\displaystyle \left(\frac13\right)^{-1} = 3$
$\displaystyle \left(-\frac13\right)^{-1} = -3$

Practice

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