5.8 Sample test: exponents

$x^{-6}$

$x^0$

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

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

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

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

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

$-2^2$

$-2^3$

$-2^{-1}$

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

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

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

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

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

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

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

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

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

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

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

$8^x=64$

$8^x=8$

$8^x=1$

$8^x=2$

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

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

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

$8^x=0$

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

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

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

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

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?