This algebra 2 radicals test answer key gives the full solution and the scoring guide for each of the 8 test questions on radicals and rational exponents.
This key gives a complete solution and a scoring guide for every question. Award partial credit as described, and give full credit for any valid method that reaches the correct result, including equivalent radical forms.
Algebra 2 Radicals Test Answer Key: questions 1–5
Question 1: Roots and powers (2 points)
- \((-5)^{3}=-125\): \(-5\).
- \((\sqrt[4]{81})^{3}=3^{3}=\) \(27\).
Scoring: 1 point for each part.
Question 2: Radical and exponent forms (2 points)
- \(\sqrt[3]{x^{5}}\), which equals \((\sqrt[3]{x})^{5}\).
- \(y^{3/4}\).
Scoring: 1 point for each part. Accept \((\sqrt[3]{x})^{5}\) in part a.
Question 3: Simplify radicals (2 points)
- \(c=\sqrt{12^{2}+6^{2}}=\sqrt{180}=\sqrt{36\cdot 5}=\) \(6\sqrt{5}\).
- \(\sqrt[3]{8x^{3}\cdot 5x}=\) \(2x\sqrt[3]{5x}\).
Scoring: 1 point for each part: 0.5 for the correct setup and 0.5 for the simplified result.
Question 4: Combine radicals (2 points)
- \(10\sqrt{3}-4\sqrt{3}+5\sqrt{3}=\) \(11\sqrt{3}\).
- \(\frac{12}{2\sqrt{2}}=\frac{6}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=\) \(3\sqrt{2}\).
Scoring: 1 point for each part: 0.5 for simplifying each radical correctly and 0.5 for the final answer.
Question 5: Exponent properties (2 points)
- \(\frac{2}{3}+\frac{3}{4}=\frac{8}{12}+\frac{9}{12}=\frac{17}{12}\): \(x^{17/12}\).
- \(16^{3/4}\cdot a^{3}=8a^{3}\): \(8a^{3}\).
Scoring: 1 point for each part.
Algebra 2 Radicals Test Answer Key: questions 6–8
Question 6: Solve equations (3 points)
- \(\sqrt{3x-2}=5\), \(3x-2=25\), \(x=9\). Check: \(2(5)-4=6\). \(x=9\)
- \(x=64^{2/3}=16\). Check: \(16^{3/2}=64\). \(x=16\)
- \(x+5=(-2)^{3}=-8\). \(x=-13\)
Scoring: 1 point for each part. Do not award the point in part a if the answer was not checked.
Question 7: Extraneous solution (3 points)
- The graphs cross once, at about \(x=3\).
- \(x+1=x^{2}-2x+1\), so \(x^{2}-3x=0\) and \(x(x-3)=0\). The candidates are \(x=0\) and \(x=3\).
- At \(x=0\): \(\sqrt{1}=1\) but \(0-1=-1\), so it fails. At \(x=3\): \(\sqrt{4}=2=3-1\). The solution is \(x=3\) and \(x=0\) is extraneous.
Scoring: 1 point for each part. Part c needs both checks for the point.
Question 8: Cube-shaped box (4 points)
- \(e=\sqrt[3]{192}=\sqrt[3]{64\cdot 3}=\) \(4\sqrt[3]{3}\) inches.
- \(4(1.44225)\approx\) \(5.77\) inches.
- \(6e^{2}=6\cdot 16\sqrt[3]{9}=\) \(96\sqrt[3]{9}\text{ in}^{2}\), about \(199.69\text{ in}^{2}\).
- Yes: \(5.77 < 6\), so the edge is shorter than the shelf height.
Scoring: 1 point for each part. Part d needs the comparison and the conclusion.
Scoring summary
20 points: add the points of every question of this algebra 2 radicals test answer key.
How to use this algebra 2 radicals test answer key
Use this algebra 2 radicals test answer key to score the test fairly and to see which skills need more practice.
Printable PDF · Test Answer Key
Algebra 2 Radicals and Rational Exponents test answer key
Download the PDF
See every chapter on the Algebra 2 math page.
