SAT Exponents and Radicals Practice Flashcards

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

Answer: 32

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.

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

Answer: x²y²/(x² + y²)

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.

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

Answer: x = 4 only

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.

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

Answer: 8

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.

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

Answer: 3x²y³

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.

If 2^x · 4^(x+1) = 8³, what is the value of x?

Answer: 7/3

Convert all bases to base 2: 4 = 2², so 4^(x+1) = 2^(2x+2). Also 8³ = (2³)³ = 2⁹. The equation becomes 2^x · 2^(2x+2) = 2⁹, which simplifies to 2^(3x+2) = 2⁹. Setting exponents equal: 3x + 2 = 9, so 3x = 7, x = 7/3. Choice A gives 3(2)+2 = 8 ≠ 9. Choice B gives 3(8/3)+2 = 10 ≠ 9. Choice C gives 3(3)+2 = 11 ≠ 9.