Bluebook SAT Test SAT Exponents and Radicals Practice 6

Question 1

If x^(2/3) = 4, what is the value of x^(5/3)?

Accepted Answer

32

Answer

Since x^(2/3) = 4, we can find x^(1/3) = √4 = 2, so x = 2³ = 8. Then x^(5/3) = (x^(1/3))^5 = 2^5 = 32. Alternatively, x^(5/3) = x^(2/3) · x^(3/3) = 4 · 8 = 32. Choice A (8) is the value of x itself — a common trap. Choice B (16) comes from squaring x^(2/3) incorrectly. Choice D (64) arises from computing 4^3.

Question 2

Which expression is equivalent to (x⁻² + y⁻²)⁻¹?

Accepted Answer

x²y²/(x² + y²)

Answer

Rewrite x⁻² + y⁻² as (1/x²) + (1/y²) = (y² + x²)/(x²y²). Taking the reciprocal gives x²y²/(x² + y²). Choice A is the most common mistake — students apply the outer exponent of −1 term-by-term, incorrectly flipping each piece and adding. Choice C confuses the structure with (x + y)⁻². Choice D omits the numerator x²y².

Question 3

How many real values of x satisfy the equation √(x + 5) = x − 1?

Accepted Answer

x = 4 only

Answer

Squaring both sides: x + 5 = (x − 1)² = x² − 2x + 1, giving x² − 3x − 4 = 0, so (x − 4)(x + 1) = 0. Candidates are x = 4 and x = −1. Check x = 4: √9 = 3 and 4 − 1 = 3 ✓. Check x = −1: √4 = 2 but −1 − 1 = −2 ✗. Because a square root cannot equal a negative number, x = −1 is extraneous. Only x = 4 is valid.

Question 4

If 9^x = 4, what is the value of 27^x?

Accepted Answer

8

Answer

Write everything in base 3: 9^x = (3²)^x = 3^(2x) = 4. Now 27^x = (3³)^x = 3^(3x) = (3^(2x))^(3/2) = 4^(3/2) = (√4)³ = 2³ = 8. Choice B (64) is the trap of computing 4³. Choice C (12) comes from multiplying 9^x · 3 = 4 · 3. Choice A (6) has no algebraic basis here.

Question 5

Which of the following is equal to ⁴√(81x⁸y¹²)?

Accepted Answer

3x²y³

Answer

Apply the fourth root to each factor: ⁴√81 = ⁴√(3⁴) = 3; ⁴√(x⁸) = x^(8/4) = x²; ⁴√(y¹²) = y^(12/4) = y³. The result is 3x²y³. Choice A and C use ⁴√81 = 9 (confusing ²√81 = 9 with ⁴√81 = 3). Choices A and B double the variable exponents by taking a square root instead of a fourth root.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test SAT Exponents and Radicals Practice 6