Negative Exponents

Show your work for each problem.

1.
What is the value of $2^{-3}$ ?
1. a) $-8$
2. b) $\frac{1}{8}$
3. c) $-6$
4. d) $8$
Answer: $\frac{1}{8}$
$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ . The negative exponent gives the reciprocal, so the result stays positive.
2.
Which expression is equal to $\frac{1}{5^2}$ ?
1. a) $5^{-2}$
2. b) $5^{2}$
3. c) $\frac{1}{5^{-2}}$
4. d) $-5^{2}$
Answer: $5^{-2}$
$\frac{1}{5^2} = 5^{-2}$ . Its value is $\frac{1}{25}$ , a positive number, so $-5^{2} = -25$ is wrong, and $\frac{1}{5^{-2}} = 5^{2} = 25$ is the reciprocal.
3.
Write $10^{-2}$ as a decimal.
Answer: 0.01
$10^{-2} = \frac{1}{100} = 0.01$ , one hundredth.
4.
Evaluate $4^{-1}$ . Write the answer as a fraction.
Answer: 1/4
$4^{-1} = \frac{1}{4^1} = \frac{1}{4}$ , the reciprocal of $4$ .
5.
Which expression is equal to $\frac{x^{-3}}{y^{-2}}$ ?
1. a) $\frac{y^2}{x^3}$
2. b) $\frac{x^3}{y^2}$
3. c) $-\frac{y^2}{x^3}$
4. d) $x^{3}y^{2}$
Answer: $\frac{y^2}{x^3}$
$x^{-3}$ moves down and becomes $x^3$ , $y^{-2}$ moves up and becomes $y^2$ : $\frac{x^{-3}}{y^{-2}} = \frac{y^2}{x^3}$ .
6.
Evaluate $3^{-2}$ . Write the answer as a fraction.
Answer: 1/9
1. Rewrite with a positive exponent: $3^{-2} = \frac{1}{3^{\square}}$ . What goes in the box? 2
2. What is $3^2$ ? 9
3. Write $3^{-2}$ as a fraction. 1/9
7.
Write $0.001$ as a power of 10. What is the exponent?
Answer: -3
1. $0.001 = \frac{1}{\square}$ . What goes in the box? 1000
2. $1000 = 10^{\square}$ . What is the exponent? 3
3. So $0.001 = \frac{1}{10^3} = 10^{\square}$ . What is the exponent? -3
8.
Which number is the largest?
1. a) $10^{-2}$
2. b) $10^{-1}$
3. c) $2^{-1}$
4. d) $2^{-3}$
Answer: $2^{-1}$
$2^{-1} = 0.5$ , $2^{-3} = 0.125$ , $10^{-1} = 0.1$ , $10^{-2} = 0.01$ . The largest is $2^{-1}$ .
9.
Evaluate $\left(\frac{2}{5}\right)^{-2}$ . Write the answer as a fraction.
Answer: 25/4
$\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2} = \frac{25}{4}$ .
10.
Evaluate $\frac{1}{2^{-4}}$ .
Answer: 16
Moving $2^{-4}$ up to the numerator changes the sign of the exponent: $\frac{1}{2^{-4}} = 2^4 = 16$ .
11.
Evaluate $\left(\frac{3}{4}\right)^{-3}$ . Write the answer as a fraction.
Answer: 64/27
1. Flip the fraction: what is the reciprocal of $\frac{3}{4}$ ? 4/3
2. What is $4^3$ ? 64
3. What is $3^3$ ? 27
4. Write $\left(\frac{4}{3}\right)^{3}$ as a fraction. 64/27
12.
Simplify $x^{-5} \cdot x^{2}$ and write it as $\frac{1}{x^{\square}}$ . What goes in the box?
Answer: 3
1. Product rule: what is $-5 + 2$ ? -3
2. $x^{-3} = \frac{1}{x^{\square}}$ . What goes in the box? 3
13.
Simplify $\frac{4a^{-3}b^{2}}{2a^{2}b^{-1}}$ and write it with positive exponents.
1. a) $\frac{2b}{a}$
2. b) $2a^{5}b^{3}$
3. c) $\frac{b^{3}}{2a^{5}}$
4. d) $\frac{2b^{3}}{a^{5}}$
Answer: $\frac{2b^{3}}{a^{5}}$
$4 \div 2 = 2$ , $a^{-3-2} = a^{-5}$ , $b^{2-(-1)} = b^{3}$ . So the result is $2a^{-5}b^{3} = \frac{2b^{3}}{a^{5}}$ ; the coefficient $2$ stays in the numerator.
14.
Evaluate $2x^{-3}$ for $x = 2$ . Write the answer as a fraction.
Answer: 1/4
1. The exponent belongs only to $x$ . What is $2^{-3}$ ? 1/8
2. Multiply by the coefficient: what is $2 \times \frac{1}{8}$ ? 1/4
15.
Evaluate $2^{-2} + 2^{-3}$ . Write the answer as a fraction.
Answer: 3/8
$2^{-2} + 2^{-3} = \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}$ . The exponents are not added here: that rule is for multiplying.
16.
Simplify $\frac{(x^{-2})^{3}}{x^{-10}}$ to a single power $x^{\square}$ . What is the exponent?
Answer: 4
1. Power of a power: what is $-2 \times 3$ ? -6
2. Quotient rule: what is $-6 - (-10)$ ? 4
17.
Evaluate $\frac{2^{-3} \cdot 2^{5}}{2^{-1}}$ .
Answer: 8
1. Numerator, product rule: what is $-3 + 5$ ? 2
2. Quotient rule: what is $2 - (-1)$ ? 3
3. What is $2^3$ ? 8
18.
Multiply $(4 \times 10^{-3}) \times (2 \times 10^{5})$ and write the result as an ordinary number.
Answer: 800
1. Multiply the numbers in front: what is $4 \times 2$ ? 8
2. Add the exponents of 10: what is $-3 + 5$ ? 2
3. The result is $8 \times 10^{2}$ . Write it as an ordinary number. 800