# 5.5 Homework: fractional exponents

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This module provides practice problems designed to develop concepts related to fractional exponents.

We have come up with the following definitions.

• $x^{0}=1$
• $x^{-a}=\frac{1}{{x}^{a}}$
• $x^{\left(\frac{a}{b}\right)}=\sqrt[b]{{x}^{a}}$

Let’s get a bit of practice using these definitions.

$100^{\left(\frac{1}{2}\right)}$

$100^{-2}$

$100^{\left(\frac{-1}{2}\right)}$

$100^{\left(\frac{3}{2}\right)}$

$100^{\left(\frac{-3}{2}\right)}$

Check all of your answers above on your calculator. If any of them did not come out right, figure out what went wrong, and fix it!

Solve for $x$ : $\frac{{x}^{}}{{x}^{}}17^{\left(\frac{1}{2}\right)}$ $17^{\left(\frac{1}{2}\right)}$

Solve for $x$ : $x^{\left(\frac{1}{2}\right)}=9$

Simplify: $\frac{x}{\sqrt{x}}$

Simplify: $\frac{{x}^{}+\sqrt{x}}{{x}^{}+\frac{1}{\sqrt{x}}}$

Multiply the top and bottom by $x^{\left(\frac{1}{2}\right)}$ .

Now…remember inverse functions? You find them by switching the $x$ and the $y$ and then solving for $y$ . Find the inverse of each of the following functions. To do this, in some cases, you will have to rewrite the things. For instance, in #9, you will start by writing $y=x^{\left(\frac{1}{2}\right)}$ . Switch the $x$ and the $y$ , and you get $x=y^{\left(\frac{1}{2}\right)}$ . Now what? Well, remember what that means: it means $x=\sqrt{y}$ . Once you’ve done that, you can solve for $y$ , right?

$x^{3}$

• Find the inverse function.
• Test it.

$x^{-2}$

• Find the inverse function.
• Test it.

$x^{0}$

• Find the inverse function.
• Test it.

Can you find a generalization about the inverse function of an exponent?

Graph $y=2^{x}$ by plotting points. Make sure to include both positive and negative $x$ values.

Graph $y=22^{x}$ by doubling all the y-values in the graph of $y=2^{x}$ .

Graph $y=2^{x}+1$ by taking the graph $y=2^{x}$ and “shifting” it to the left by one.

Graph $y=\left(\frac{1}{2}\right)^{x}$ by plotting points. Make sure to include both positive and negative $x$ values.

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