Question

1. $\frac{1}{x^3} = x^{-3}$
2. $\frac{1}{a^2} = a^{-2}$
3. $\frac{1}{49} = $
4. $\frac{1}{8} = $
5. $\frac{1}{144} = $
6. $\frac{1}{169} = 13^{-2}$

evaluate each expression if $x = 3$, $y = -2$, and $z = 4$.

1. $x^{-4} = \frac{1}{81}$
2. $y^{-2} = \frac{1}{4}$
3. $y^{-5} = \frac{1}{32}$
4. $z^{-4} = \frac{1}{256}$
5. $5^{-2} = \frac{1}{25}$
6. $10^{-2} = $
7. $3z^{-1} = $
8. $z^4 = \frac{1}{16}$
9. $(xz)^{-2} = $
10. hair grows at a rate of $\frac{1}{64}$ inch per day. write this number using negative exponents.

Explanation:

Step1: Apply negative exponent rule

Recall the rule $\frac{1}{a^n}=a^{-n}$ for non-zero $a$.

Problem 21:

$\frac{1}{49}=\frac{1}{7^2}=7^{-2}$

Problem 22:

$\frac{1}{8}=\frac{1}{2^3}=2^{-3}$

Problem 23:

$\frac{1}{144}=\frac{1}{12^2}=12^{-2}$

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Step2: Substitute values for variables

Use $x=3, y=-2, z=4$ and $a^{-n}=\frac{1}{a^n}$.

Problem 25:

$x^{-4}=3^{-4}=\frac{1}{3^4}=\frac{1}{81}$

Problem 26:

$y^{-2}=(-2)^{-2}=\frac{1}{(-2)^2}=\frac{1}{4}$

Problem 27:

$y^{-5}=(-2)^{-5}=\frac{1}{(-2)^5}=-\frac{1}{32}$

Problem 28:

$z^{-4}=4^{-4}=\frac{1}{4^4}=\frac{1}{256}$

Problem 29:

$5^2=5\times5=25$

Problem 30:

$10^7=10\times10\times10\times10\times10\times10\times10=10000000$

Problem 31:

$3z^{-1}=3\times4^{-1}=3\times\frac{1}{4}=\frac{3}{4}$

Problem 32:

$x^2=3^2=3\times3=9$

Problem 33:

$(xz)^{-4}=(3\times4)^{-4}=12^{-4}=\frac{1}{12^4}=\frac{1}{20736}$

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Step3: Rewrite fraction with negative exponent

Problem 34:

$\frac{1}{64}=\frac{1}{2^6}=2^{-6}$

Answer:

1. $7^{-2}$
2. $2^{-3}$
3. $12^{-2}$
4. $\frac{1}{81}$
5. $\frac{1}{4}$
6. $-\frac{1}{32}$
7. $\frac{1}{256}$
8. $25$
9. $10000000$
10. $\frac{3}{4}$
11. $9$
12. $\frac{1}{20736}$
13. $2^{-6}$