← Back to the Rational Exponents test
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Each solution shows the work and a scoring guide. Give partial credit for correct setup, and remember that a candidate answer that is not checked earns no credit for the final answer in the radical-equation questions.
Standards: N-RN.A.1, N-RN.A.2, A-REI.A.2 · Level: Grade 11 Algebra 2
Question 1 Evaluate (2 points)
a) (−6)3 = −216: −6.
b) (5√32)3 = 23 = 8.
c) 81−3/4 = 1/(4√(81))3 = 1/27: 127.
d) 54 = 625: 5.
Scoring: 0.5 point each.
Question 2 Changing forms (2 points)
a) x3/5.
b) 4√(y7) or (4√(y))7.
Scoring: 1 point each.
Question 3 Simplify radicals (3 points)
a) 98x6 = 49x6 · 2: 7x3√2.
b) 81y5 = 27y3 · 3y2: 3y∛(3y2).
Scoring: 1 point for (a); 1 point for the factor 3y, 1 point for ∛(3y2).
Question 4 Exponent rules (2 points)
a) Exponent: 3/2 + 1/4 − (−1/4) = 2. x2.
b) 163/4 = 8 and (4/3)(3/4) = 1: 8a.
Scoring: 1 point each.
Question 5 Radical expressions (3 points)
a) 3 · 2√5 + 3√5 = 9√5.
b) Multiply by 3 + √2: 10(3 + √2)9 − 2 = 30 + 10√27.
Scoring: 1 point for (a); 1 point for the conjugate step, 1 point for the result.
Question 6 Radical equation and its graph (3 points)
a) 5x + 1 = x2 − 2x + 1, so x2 − 7x = 0 and x = 0 or x = 7. Check 0: √1 = 1 but 0 − 1 = −1 ✗. Check 7: √36 = 6 = 7 − 1 ✓. x = 7
b) The graphs meet only at (7, 6); at x = 0 the line is below the x-axis while the radical is positive, so 0 is extraneous.
Scoring: 1 point for the squared equation and candidates, 1 for the check; 1 point for the explanation.
Question 7 Two radicals (3 points)
√(x + 3) = 3 − √x. Square: x + 3 = 9 − 6√x + x, so 6√x = 6 and x = 1.
Check: √4 + √1 = 2 + 1 = 3 ✓.
x = 1
Scoring: 1 point for isolating and squaring, 1 for solving, 1 for the check.
Question 8 Falling time (2 points)
a) √h / 4 = 3.5, so √h = 14 and h = 196. 196 feet
b) √144 / 4 = 12/4 = 3 seconds.
Scoring: 1 point each.