ACT Math: Number and Quantity (Drill 2)

Question 1. Which of the following is equivalent to x⁻³ · x⁵?

• A) x⁻¹⁵
• B) x² ✓
• C) x⁸
• D) 1/x²

Explanation: When multiplying expressions with the same base, add the exponents: x⁻³ · x⁵ = x^{−3 + 5} = x². Choice A results from multiplying the exponents (−3 × 5 = −15), which is the rule for a power raised to a power, not a product. Choice C results from adding the absolute values (3 + 5 = 8) and ignoring the negative sign. Choice D would be correct only if the exponents summed to −2, but −3 + 5 = 2, not −2.

Question 2. Which of the following is equivalent to 8^2/3?

• A) 4 ✓
• B) 16/3
• C) √8²
• D) 2/3

Explanation: A rational exponent a^m/n means the nth root of a raised to the m power: 8^2/3 = (∛8)² = 2² = 4, since the cube root of 8 is 2. Choice B results from multiplying 8 × (2/3) = 16/3. Choice C shows a square root instead of a cube root, misreading the denominator of the exponent as 2 rather than 3. Choice D confuses the value of the exponent itself with the answer.

Question 3. Which of the following matrices is equal to −3 multiplied by the matrix [[2, −1], [0, 4]]?

• A) [[−6, 3], [0, −12]] ✓
• B) [[6, −3], [0, 12]]
• C) [[−6, −1], [0, 4]]
• D) [[−1, 2], [3, −7]] as given in the problem

Explanation: Scalar multiplication means multiplying every entry of the matrix by the scalar: −3 × 2 = −6; −3 × (−1) = 3; −3 × 0 = 0; −3 × 4 = −12. The result is [[−6, 3], [0, −12]]. Choice B shows all signs flipped from the correct answer; it is the result of multiplying by +3 instead of −3. Choice C only applies the scalar to the first column and leaves the second column unchanged. Choice D results from adding −3 to each entry rather than multiplying.

Question 4. For all positive values of x, which of the following is equivalent to ⁵√x³?

• A) x^5/3
• B) x^3/5 ✓
• C) x^2/5
• D) x¹⁵

Explanation: The nth root of x^m equals x^m/n, the power is the numerator and the root index is the denominator. So ⁵√x³ = x^3/5. Choice A flips the fraction, placing the root index in the numerator and the power in the denominator. Choice C puts 5 − 3 = 2 in the numerator instead of the power 3. Choice D multiplies 3 × 5 = 15 rather than forming a fraction.

Question 5. What is the sum of the matrices [[3, −2], [5, 1]] and [[−1, 4], [−5, 2]]?

• A) [[2, 2], [0, 3]] ✓
• B) [[4, 2], [0, −1]]
• C) [[2, −6], [10, 2]]
• D) [[2, 2], [0, −1]]

Explanation: Add corresponding entries: top-left: 3 + (−1) = 2; top-right: −2 + 4 = 2; bottom-left: 5 + (−5) = 0; bottom-right: 1 + 2 = 3. The sum is [[2, 2], [0, 3]]. Choice B subtracts the second matrix from the first in the top-left entry: 3 − (−1) = 4. Choice C subtracts the second matrix from the first throughout. Choice D gets the top row correct but makes a sign error in the bottom-right: 1 − 2 = −1 instead of 1 + 2 = 3.