Nth Roots

Show your work for each problem.

1.
What is ?
1. a)3
2. b)27
3. c)9
4. d)81
Answer: 3
because .
2.
Evaluate
Answer: 2
because .
3.
Which expression is equivalent to ?
1. a)
2. b)
3. c)
4. d)
Answer:
, so . The index becomes the denominator of the exponent.
4.
Evaluate
Answer: 2
because . We can write .
5.
Evaluate
Answer: 2
1. Express 64 as a power of 2. What is 6
2. So we have . What is the answer? 2
6.
Which statement is TRUE about ?
1. a)It equals
2. b)It equals
3. c)It equals
4. d)It is not a real number
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 .
7.
Evaluate
Answer: -3
because . Odd roots of negative numbers exist because an odd number of negative factors gives a negative product.
8.
Convert to fractional exponent form
Answer: x^(3/4)
1. Using , what goes in the numerator? 3
2. What goes in the denominator? 4
3. Write the final answer as to what power? 3/4
9.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
. The exponent 6 divided by the index 3 equals 2.
10.
Simplify
Answer: 2*4throot(5)
1. Factor 80 to find a perfect 4th power: 5
2. We have . What is ? 2
3. The simplified form is 5
11.
Evaluate
Answer: 3
because . We can verify: .
12.
Simplify
Answer: x^(3/2)*y^2
1. Apply the rule to : What is simplified? 3/2
2. Apply the rule to : What is ? 2
3. The answer is 2
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 . This is known as the Rule of 72: divide 72 by the rate to get doubling time, or divide 72 by years to get the rate.
14.
Simplify
Answer: -5x^2
1. First, can we take cube root of a negative? (yes/no) yes
2. What is ? -5
3. What is ? x^2
4. Combine: The answer is? -5x^2
15.
Simplify
Answer: 4
, so . Alternatively, using exponent rules: .
16.
Find the geometric mean of 4, 8, and 16
Answer: 8
1. The geometric mean uses which root for 3 numbers? cube root
2. Calculate the product: 512
3. What is ? 8