ACT · Math · Preparing for Higher Math: Number and Quantity

Exponents and radicals

Exponents and radicals is part of Preparing for Higher Math: Number and Quantity in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include product, quotient, and power rules, negative and zero exponents, simplify a radical and rationalize a denominator.

Common traps: multiplying exponents when multiplying powers; adding exponents in a power of a power; applying the exponent to only one factor.

Sample questions

Question 1
Which of the following is equivalent to (2x³)⁴?
- A.
  2x¹²
- B.
  8x⁷
- C.
  8x¹²
- D.
  16x¹²
Answer: D. 16x¹²
Raise each factor to the 4th power: 2⁴ = 16 and (x³)⁴ = x^(3 · 4) = x¹², giving 16x¹². Choice A applies the exponent only to x; B multiplies 2 by 4 and adds the exponents 3 + 4; C multiplies 2 by 4 instead of raising 2 to the 4th power.
Question 2
What value of x satisfies 3^(x + 2) = 81?
- A.
  2
- B.
  4
- C.
  25
- D.
  79
Answer: A. 2
Write 81 as a power of 3: 81 = 3⁴, so x + 2 = 4 and x = 2. Choice B stops at the value of the exponent x + 2; C divides 81 by 3 and then subtracts 2, treating the exponent as a factor; D subtracts 2 from 81 as if the exponent could be removed.
Question 3
For x > 0, which of the following is equivalent to ∛(54x⁴)?
- A.
  3x∛(2x)
- B.
  3x∛(6x)
- C.
  3x²∛2
- D.
  9x∛(2x)
Answer: A. 3x∛(2x)
Pull out perfect cubes: 54 = 27 · 2 and x⁴ = x³ · x, so ∛(54x⁴) = ∛27 · ∛(x³) · ∛(2x) = 3x∛(2x). Choice B factors 54 as 9 · 6 and takes the square root of 9, but 9 is not a perfect cube; C halves the exponent 4 as a square root would; D takes ∛27 to be 9.
Question 4
Which of the following is equivalent to 3√8 − √50 + √18?
- A.
  −2√2
- B.
  0
- C.
  4√2
- D.
  14√2
Answer: C. 4√2
Simplify each term: 3√8 = 6√2, √50 = 5√2 and √18 = 3√2. Then 6√2 − 5√2 + 3√2 = 4√2. Choice A subtracts √18 as well; B simplifies 3√8 to 2√2, dropping the 3; D adds all three terms, ignoring the minus sign.
Question 5
For x > 1, (x^(3n) · x^(n + 2))/x^(2n − 4) = x²⁰. What is the value of n?
- A.
  11/3
- B.
  7
- C.
  9
- D.
  11
Answer: B. 7
Combine the exponents: 3n + (n + 2) − (2n − 4) = 2n + 6, so 2n + 6 = 20 and n = 7. Choice A adds the denominator's exponent, 6n − 2 = 20; C drops the −4, solving 2n + 2 = 20; D subtracts 4 instead of adding it, solving 2n − 2 = 20.

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