SAT Math: Rational Exponents and Radical Expressions Practice Questions
Connect fractional exponents with roots, simplify radical expressions, and apply exponent rules on their stated domains. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of SAT.
Answer the questions below — you get instant feedback and a full explanation for each.
1. What is the value of 64^(2/3)?
Explanation. The denominator 3 indicates a cube root, and the numerator 2 indicates a square. The cube root of 64 is 4, so 64^(2/3) = 4^2 = 16.
2. A positive number a satisfies a^(3/2) = 27. What is a?
Explanation. For positive a, a^(3/2) = (sqrt(a))^3. Thus sqrt(a) = 3 and a = 9. Substitution checks: 9^(3/2) = 3^3 = 27.
3. What is the value of 16^(3/4)?
Explanation. Take the fourth root of 16, which is 2, and then cube it: 16^(3/4) = 2^3 = 8.
4. For positive x, x^(1/3) times x^(2/3) equals x^k. What is k?
Explanation. When multiplying powers with the same positive base, add the exponents: 1/3 + 2/3 = 1. Therefore k = 1.
5. The expression sqrt(75) is written as a sqrt(3), where a is positive. What is a?
Explanation. Factor 75 as 25 times 3. Then sqrt(75) = sqrt(25)sqrt(3) = 5sqrt(3), so a = 5.
6. For every real x, sqrt(81x^4) = kx^2, where sqrt denotes the principal square root. What is k?
Explanation. Since x^2 is nonnegative, the principal square root of x^4 is x^2. Also sqrt(81) = 9, so sqrt(81x^4) = 9x^2 and k = 9.
📘 Looking for more study material? Browse SAT prep books and study resources. Some links are affiliate links; we may earn a commission at no cost to you.
FAQ
What does a rational exponent mean?
For a positive base, a^(m/n) is the nth root of a raised to the mth power. A positive base avoids ambiguity about real roots and exponent rules.
Why do square roots sometimes require absolute values?
A principal square root is nonnegative. Thus sqrt(x^2) equals |x|. Since x^2 is already nonnegative, sqrt(x^4) equals x^2.