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 equal to √50 in simplest radical form?
- A.
  5√2
- B.
  10√2
- C.
  5√10
- D.
  25√2
Answer: A. 5√2
√50 = √(25 · 2) = √25 · √2 = 5√2. Choice B (10√2) doubles the coefficient instead of using √25 = 5. Choice C (5√10) splits 50 as 5 × 10 and leaves 10 under the radical. Choice D (25√2) pulls 25 out of the radical without taking its square root.
Question 2
What value of x satisfies √(x + 6) = 5?
- A.
  −1
- B.
  19
- C.
  25
- D.
  31
Answer: B. 19
Square both sides: x + 6 = 25, so x = 19. Check: √(19 + 6) = √25 = 5. Choice A subtracts 6 from 5 without squaring; C squares 5 but forgets to subtract 6; D adds 6 instead of subtracting it.
Question 3
A drone service in Singapore recorded a total of 6 × 10⁸ centimeters flown, shared equally among 3 × 10⁴ deliveries. How many centimeters did the drones fly per delivery, in scientific notation?
- A.
  2 × 10²
- B.
  2 × 10⁴
- C.
  2 × 10¹²
- D.
  2 × 10³²
Answer: B. 2 × 10⁴
Divide the coefficients, 6 ÷ 3 = 2, and subtract the exponents, 8 − 4 = 4. The result is 2 × 10⁴ cm per delivery. Choice A (2 × 10²) divides the exponents, 8 ÷ 4. Choice C (2 × 10¹²) adds the exponents instead of subtracting. Choice D (2 × 10³²) multiplies the exponents.
Question 4
Which expression equals 6/(√3 + 1)?
- A.
  3√3 − 3
- B.
  6√3 − 6
- C.
  3√3 + 3
- D.
  6√3 + 6
Answer: A. 3√3 − 3
A is right: Multiply by (√3 − 1)/(√3 − 1): 6(√3 − 1)/(3 − 1) = 3(√3 − 1) = 3√3 − 3. B forgets to divide by 3 − 1 = 2. C multiplies by √3 + 1 instead of the conjugate. D makes both errors.
Question 5
A shipping firm models the number of box layers with the equation 3^(2x − 1) = 81. What is the value of x?
- A.
  2.5
- B.
  4
- C.
  5
- D.
  41
Answer: A. 2.5
Since 81 = 3⁴, the exponents match: 2x − 1 = 4, so x = 2.5. B stops at the exponent value 4; C gives 2x, not x; D sets the exponent equal to 81 instead of to 4.

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