Algebra 2 Radicals and Rational Exponents Test Solutions and Scoring (PDF)
This rational exponents test answers for Algebra 2 covers radicals and rational exponents (Common Core N-RN.A.1, N-RN.A.2), with a free PDF to print.
Algebra 2 · Common Core State Standards: N-RN.A.1, N-RN.A.2
Rational exponents test answers: worked solutions
Question 1 – Convert forms [2 pts]
- \(x^{2/5}\)
- \(\sqrt[3]{7}\)
(a) \(x^{2/5}\); (b) \(\sqrt[3]{7}\)
Scoring: 1 point for each conversion.
Question 2 – Evaluate [3 pts]
-
\(\sqrt[3]{64} = 4\)
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\(\sqrt[5]{32} = 2\), \(2^{3} = 8\)
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\(4^{-3/2} = \frac{1}{(\sqrt{4})^{3}} = \frac{1}{8}\)
(a) \(4\); (b) \(8\); (c) \(\frac{1}{8}\)
Scoring: 1 point for each part.
Question 3 – Variable expressions [2 pts]
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\(\frac{3}{5} + \frac{7}{5} = 2\), so \(x^{2}\)
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\(27^{1/3} = 3\), \((a^{6})^{1/3} = a^{2}\), so \(3a^{2}\)
(a) \(x^{2}\); (b) \(3a^{2}\)
Scoring: 1 point for each part.
Question 4 – Quotient [2 pts]
\(x^{7/3 – 1/3} = x^{2}\) and \(y^{-1/2 – 3/2} = y^{-2}\).
\(\frac{x^{2}}{y^{2}}\)
Scoring: 1 point for each variable.
Question 5 – Solve [3 pts]
\((\sqrt[3]{x})^{2} = 25\), so \(\sqrt[3]{x} = 5\) or \(-5\).
Cube: \(x = 125\) or \(x = -125\).
Check: \(125^{2/3} = 25\) and \((-125)^{2/3} = 25\).
\(x = 125\) or \(x = -125\)
Scoring: 1 point for isolating the cube root with both signs, 1 point for both cubes, 1 point for the checks.
Question 6 – Animal metabolism [2 pts]
\(1^{3/4} = 1\); \(16^{3/4} = 8\); \(81^{3/4} = (\sqrt[4]{81})^{3} = 27\).
Multiply by 70: 70, 560, 1,890.
70; 560; 1,890 kcal per day
Scoring: 1 point for the m^(3/4) column, 1 point for the R column.
Question 7 – Find the error [2 pts]
In \(8^{2/3}\) the denominator 3 is the root index and the numerator 2 is the power. Liam swapped them.
Correct: \((\sqrt[3]{8})^{2} = 2^{2} = 4\).
The index is 3, not 2; the value is 4
Scoring: 1 point for identifying the swap, 1 point for the value.
Question 8 – Kepler's law [2 pts]
\(a^{3/2} = 125\). Raise both sides to \(2/3\): \(a = 125^{2/3} = 5^{2} = 25\).
Check: \(25^{3/2} = 5^{3} = 125\).
25 astronomical units
Scoring: 1 point for the reciprocal power, 1 point for the value.
Question 9 – Growth factor [2 pts]
After 6 hours: \((3^{1/2})^{6} = 3^{3} = 27\).
After 5 hours: \((3^{1/2})^{5} = 3^{5/2} = 9\sqrt{3}\).
27 and \(9\sqrt{3}\)
Scoring: 1 point for each multiplier.
Total score
20 points
Key facts behind this rational exponents test answers
- a^(1/n) is the nth root of a.
- a^(m/n) = (nth root of a)^m: root index = denominator, power = numerator.
- a^(-p) = 1 / a^p, also for fractional p.
- Product, quotient and power rules work by adding, subtracting and multiplying exponents.
- Even roots of negative numbers are not real.
How to use this rational exponents test answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
All Algebra 2 chapters: Algebra 2 math.
