Nth Roots

Show your work for each problem.

1.
What is $\sqrt[3]{27}$ ?
1. a)9
2. b)27
3. c)3
4. d)81
Answer: 3
$\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 3^3 = 27$ .
2.
Evaluate $\sqrt[4]{16}$
Answer: 2
$\sqrt[4]{16} = 2$ because $2^4 = 2 \times 2 \times 2 \times 2 = 16$ .
3.
Which expression is equivalent to $\sqrt[5]{a}$ ?
1. a) $a^5$
2. b) $5a$
3. c) $a - 5$
4. d) $a^{1/5}$
Answer: $a^{1/5}$
$\sqrt[n]{a} = a^{1/n}$ , so $\sqrt[5]{a} = a^{1/5}$ . The index becomes the denominator of the exponent.
4.
Evaluate $\sqrt[5]{32}$
Answer: 2
$\sqrt[5]{32} = 2$ because $2^5 = 32$ . We can write $32 = 2 \times 2 \times 2 \times 2 \times 2$ .
5.
Evaluate $\sqrt[6]{64}$
Answer: 2
1. Express 64 as a power of 2. What is $64 = 2^{\square}$ 6
2. So we have $\sqrt[6]{2^6}$ . What is the answer? 2
6.
Which statement is TRUE about $\sqrt[4]{-81}$ ?
1. a)It equals $3$
2. b)It is not a real number
3. c)It equals $-81$
4. d)It equals $-3$
Answer: It is not a real number
Even roots of negative numbers don't exist in real numbers. Any real number raised to an even power is positive, so no real number satisfies $x^4 = -81$ .
7.
Evaluate $\sqrt[5]{-243}$
Answer: -3
$\sqrt[5]{-243} = -3$ because $(-3)^5 = -243$ . Odd roots of negative numbers exist because an odd number of negative factors gives a negative product.
8.
Convert $\sqrt[4]{x^3}$ to fractional exponent form
Answer: $x^{3/4}$
1. Using $\sqrt[n]{a^m} = a^{m/n}$ , what goes in the numerator? 3
2. What goes in the denominator? 4
3. Write the final answer as $x$ to what power? 3/4
9.
Simplify $\sqrt[3]{x^6}$
1. a) $x^{1/2}$
2. b) $x^{18}$
3. c) $x^3$
4. d) $x^2$
Answer: $x^2$
$\sqrt[3]{x^6} = x^{6/3} = x^2$ . The exponent 6 divided by the index 3 equals 2.
10.
Simplify $\sqrt[4]{80}$
Answer: $2\sqrt[4]{5}$
1. Factor 80 to find a perfect 4th power: $80 = 16 \times \square$ 5
2. We have $\sqrt[4]{16 \times 5}$ . What is $\sqrt[4]{16}$ ? 2
3. The simplified form is $2\sqrt[4]{\square}$ 5
11.
Evaluate $\sqrt[4]{625}$
Answer: 5
$\sqrt[4]{625} = 5$ because $5^4 = 625$ . We can verify: $5 \times 5 \times 5 \times 5 = 25 \times 25 = 625$ .
12.
Simplify $\sqrt[6]{x^9y^{12}}$
Answer: $x^{3/2}y^2$
1. Apply the rule to $x^9$ : the exponent is $\frac{9}{6} = \frac{\square}{2}$ . What goes in the box? 3
2. Apply the rule to $y^{12}$ : What is $12/6$ ? 2
3. Rewrite $x^{3/2}$ as a radical: $x^{3/2} = \sqrt{x^{\square}}$ . What goes in the box? 3
13.
If an investment doubles in 12 years, what is the approximate annual growth rate?
1. a)About 12%
2. b)About 50%
3. c)About 8%
4. d)About 6%
Answer: About 6%
Rate $= \sqrt[12]{2} - 1 = 2^{1/12} - 1 \approx 1.0595 - 1 = 0.0595 \approx 6\%$ . The Rule of 72 gives a quick check: $72 \div 12 = 6$ , so about 6% per year.
14.
Simplify $\sqrt[3]{-125x^6}$
Answer: $-5x^2$
1. First, can we take cube root of a negative? yes
2. What is $\sqrt[3]{-125}$ ? -5
3. What is $\sqrt[3]{x^6}$ ? x^2
4. Combine: The answer is? -5x^2
15.
Simplify $\sqrt[4]{(-2)^8}$
Answer: 4
$(-2)^8 = 256$ , so $\sqrt[4]{(-2)^8} = \sqrt[4]{256} = 4$ . Alternatively, using exponent rules: $\sqrt[4]{(-2)^8} = |{-2}|^{8/4} = 2^2 = 4$ .
16.
Find the geometric mean of 4, 8, and 16
Answer: 8
1. Which root does the geometric mean of 3 numbers use? cube root
2. Calculate the product: $4 \times 8 \times 16 = ?$ 512
3. What is $\sqrt[3]{512}$ ? 8