# 5.8 Sample test: exponents

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This module provides a sample test over exponents.

$x^{-6}$

$x^{0}$

$x^{\left(\frac{1}{8}\right)}$

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

$\frac{1}{{a}^{-3}}$

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

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

$-2^{2}$

$-2^{3}$

$-2^{-1}$

$-9^{\left(\frac{1}{2}\right)}$

$-8^{\left(\frac{1}{2}\right)}$

$y^{\left(\frac{1}{4}\right)}y^{\left(\frac{3}{4}\right)}$

$\frac{{4x}^{4}{y}^{5}z}{6{\text{wxy}}^{-3}{z}^{3}}$

$x^{\left(\frac{1}{2}\right)}^{3}$

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

${x}^{\frac{6}{3}}$

$x^{\left(\frac{-3}{4}\right)}$

$(4\times 9)^{\left(\frac{1}{2}\right)}$

$4^{\left(\frac{1}{2}\right)}9^{\left(\frac{1}{2}\right)}$

Give an algebraic formula that gives the generalization for #18-19.

Solve for $x$ :

$8^{x}=64$

$8^{x}=8$

$8^{x}=1$

$8^{x}=2$

$8^{x}=\frac{1}{8}$

$8^{x}=\frac{1}{64}$

$8^{x}=\frac{1}{2}$

$8^{x}=0$

Rewrite $\frac{1}{\sqrt[3]{{x}^{2}}}$ as xsomething.

Solve for $x$ :

$2^{(x+3)}2^{(x+4)}=2$

${3}^{\left({x}^{2}\right)}$ = ${\left(\frac{1}{9}\right)}^{3x}$

A friend of yours is arguing that $\mathrm{x⅓}$ should be defined to mean something to do with “fractions, or division, or something.” You say “No, it means _____ instead.” He says “That’s a crazy definition!” Give him a convincing argument why it should mean what you said it means.

On October 1, I place 3 sheets of paper on the ground. Each day thereafter, I count the number of sheets on the ground, and add that many again. (So if there are 5 sheets, I add 5 more.) After I add my last pile on Halloween (October 31), how many sheets are there total?

• ## A

Give me the answer as a formula.
• ## B

Plug that formula into your calculator to get a number.
• ## C

If one sheet of paper is $\frac{1}{\text{250}}$ inches thick, how thick is the final pile?

## Depreciation

The Web site www.bankrate.com defines depreciation as “the decline in a car’s value over the course of its useful life” (and also as “something new-car buyers dread”). The site goes on to say:

Let’s start with some basics. Here’s a standard rule of thumb about used cars. A car loses 15 percent to 20 percent of its value each year.

For the purposes of this problem, let’s suppose you buy a new car for exactly \$10,000, and it loses only 15% of its value every year.

• ## A

How much is your car worth after the first year?
• ## B

How much is your car worth after the second year?
• ## C

How much is your car worth after the nth year?
• ## D

How much is your car worth after ten years? (This helps you understand why new-car buyers dread depreciation.)

Draw a graph of $y=23^{x}$ . Make sure to include negative and positive values of $x$ .

Draw a graph of $y=\left(\frac{1}{3}\right)^{x}-3$ . Make sure to include negative and positive values of $x$ .

What are the domain and range of the function you graphed in number 36?

## Extra credit:

We know that $(a+b)^{2}$ is not, in general, the same as $a^{2}+b^{2}$ . But under what circumstances, if any, are they the same?

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yes that's correct
Professor
I think
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