Exponents and radicals questions on Digital SAT Math almost always come down to one core move: rewriting a radical as a fractional exponent (the nth root of x equals x to the power 1/n) so that the normal rules of exponents apply. Once everything is in exponent form, multiplying, dividing, and raising powers to powers become simple addition, subtraction, and multiplication of the exponents themselves — the hard part is almost always the initial conversion and staying organized with fractions.
The 13 problems below cover this full range, including several problems that ask you to solve for an unknown exponent variable, simplify radical expressions with variables under the root, and rearrange literal equations that involve a square root. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. The fastest way through nearly all of these is to convert every radical to fractional exponent form immediately, before doing any other algebra — trying to manipulate radical symbols directly almost always takes longer and invites errors. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
√(a²b⁶). If a and b are positive numbers, which of the following is equivalent to the expression above?
Split the square root over each factor: √(a²b⁶) = √(a²) × √(b⁶).
Since a and b are positive, √(a²) = a and √(b⁶) = b³ (because (b³)² = b⁶). That gives ab³. The correct answer is B.
Which expression is equivalent to ⁾⁵√(32a³b⁴), where a > 0 and b > 0?
Rewrite the fifth root as a power of 1/5, and recognize 32 = 2⁵: ⁾⁵√(32a³b⁴) = (2⁵a³b⁴)1/5.
Distribute the exponent 1/5 to each factor: 2⁵×(1/5) = 2¹ = 2, a3/5, and b4/5. That gives 2a3/5b4/5. The correct answer is A.
The expression [x²⁰(x − 4) + 4x²⁰] / (5x²) is equivalent to (1/5)xᶜ, where c is a constant and x > 0. What is the value of c?
Factor x²⁰ out of the numerator: x²⁰(x − 4) + 4x²⁰ = x²⁰[(x − 4) + 4] = x²⁰ × x = x²&sup9;.
Now divide: x²&sup9; / (5x²) = (1/5)x²&sup9;⁻² = (1/5)x¹⁵. So c = 19. The correct answer is C.
W = R²/(R² + S²). The given equation relates the variables R, S, and W, where R, S, and W represent positive numbers. Which equation correctly expresses S in terms of R and W?
Multiply both sides by (R² + S²): W(R² + S²) = R², so WR² + WS² = R².
Isolate the S² term: WS² = R² − WR², so S² = (R² − WR²)/W. Take the square root of both sides: S = √((R² − WR²)/W). The correct answer is A.
f(x) = 9x − 7. Function f is defined by the given equation. The function g is defined by g(x) = f(x) − (7x − 2). What is the x-coordinate of the x-intercept of the graph of y = g(x) in the xy-plane?
Substitute f(x): g(x) = (9x − 7) − (7x − 2) = 9x − 7 − 7x + 2 = 2x − 5.
The x-intercept occurs where g(x) = 0: 2x − 5 = 0, so x = 5/2. The correct answer is C.
If n and k are numbers greater than 1 and the cube root of n⁴ is equivalent to the cube root of k², for what value of a is n2a+1 equal to k?
Since both sides have the same root index, the expressions under the radicals must be equal: n⁴ = k². Taking the square root of both sides (n, k > 1, so both positive): k = n².
Set n2a+1 equal to n² and match exponents: 2a + 1 = 2, so 2a = 1, giving a = 1/2.
⁵⁵√(119n) × (⁶√(119n))² is equivalent to (119n)30x, where n > 1. For what value of x is this true?
Rewrite every radical as a fractional exponent: ⁵⁵√(119n) = (119n)1/5, and (⁶√(119n))² = (119n)2/6 = (119n)1/3.
Multiply by adding exponents: (119n)1/5 + 1/3 = (119n)3/15 + 5/15 = (119n)8/15. Set 30x = 8/15, so x = 8/450 = 4/225.
The fifth root of p² equals t3/4. In the given equation, p > 1 and t > 1. If t = p2n−1, where n is a constant, what is the value of n?
Rewrite the fifth root as an exponent: p2/5 = t3/4. Substitute t = p2n−1: p2/5 = (p2n−1)3/4 = p3(2n−1)/4.
Since the bases match (p > 1), the exponents must be equal: 2/5 = 3(2n−1)/4. Cross-multiply: 8 = 15(2n−1), so 8 = 30n − 15, giving 30n = 23 and n = 23/30.
Which expression is equivalent to (¹²√(x⁵y⁵)), where x and y are positive?
Since x⁵y⁵ = (xy)⁵, the twelfth root becomes: ¹²√((xy)⁵) = (xy)5/12.
The correct answer is B.
The expression (6x²)(3x⁵) is equivalent to bx⁹, where b is a constant. What is the value of b?
Multiply the coefficients and add the exponents: (6)(3) = 18, and x² × x⁵ = x⁹.
That gives 18x⁹, so b = 18.
r = √(5w − 3). The given equation relates the real number r and w, where w > 3/5. Which equation correctly expresses w in terms of r?
Square both sides to eliminate the radical: r² = 5w − 3.
Add 3 to both sides: r² + 3 = 5w. Divide by 5: w = r²/5 + 3/5. The correct answer is C.
If n and k are numbers greater than 1 and the fifth root of n⁵ is equivalent to the fifth root of k², for what value of a is n2a+1 equal to k?
Since both sides share the same root index, the radicands must be equal: n⁵ = k². Taking the square root of both sides (n, k > 1): k = n5/2.
Set n2a+1 equal to n5/2 and match exponents: 2a + 1 = 5/2, so 2a = 3/2, giving a = 3/4.
The fifth root of xᵍ can be rewritten as the cube root of x, where m is a constant and x > 1. What is the value of m?
Rewrite both radicals as fractional exponents: (xᵍ)1/5 = x1/3, which means xm/5 = x1/3.
Since the bases match (x > 1), the exponents must be equal: m/5 = 1/3, so m = 5/3.
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