Algebra 2 Radicals and Rational Exponents Test with Point Values (PDF)

Algebra 2 math test: Radicals and Rational Exponents – Maths PDF

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This rational exponents test for Algebra 2 covers radicals and rational exponents (Common Core N-RN.A.1, N-RN.A.2), with a free PDF to print.

This 45-minute test on radicals and rational exponents has 9 questions worth 20 points in total. The questions move from direct skills to multi-step problems. Show your work, because partial credit is given for a correct setup and sound reasoning.

Time and points

About this test

Grade: Algebra 2
Chapter: Radicals and Rational Exponents
Standards: N-RN.A.1, N-RN.A.2
Time: 45 minutes
Total: 20 points

Rational exponents test: questions

Question 1 – Convert forms [2 pts]

Write \(\sqrt[5]{x^{2}}\) with a rational exponent. (1 point)
Write \(7^{1/3}\) as a radical. (1 point)

Question 2 – Evaluate [3 pts]

Evaluate each expression without a calculator.

\(64^{1/3}\) (1 point)
\(32^{3/5}\) (1 point)
\(4^{-3/2}\) (1 point)

Question 3 – Variable expressions [2 pts]

Simplify each expression.

\(x^{3/5} \cdot x^{7/5}\) (1 point)
\((27a^{6})^{1/3}\) (1 point)

Question 4 – Quotient [2 pts]

Simplify \(\frac{x^{7/3}y^{-1/2}}{x^{1/3}y^{3/2}}\) and write the answer with positive exponents.

Question 5 – Solve [3 pts]

Graphs of y = x^(1/2) and y = x^(1/3).

Solve \(x^{2/3} = 25\) for all real \(x\). Check each solution.

Question 6 – Animal metabolism [2 pts]

Fill in the table.

Metabolic rate is \(R = 70m^{3/4}\) kcal per day. Complete the table.

Question 7 – Find the error [2 pts]

Liam wrote \(8^{2/3} = \sqrt{8^{3}} = \sqrt{512}\). Explain the error and find the correct value of \(8^{2/3}\).

Question 8 – Kepler's law [2 pts]

Planet data: T = a^(3/2).

A planet has period \(T = a^{3/2}\) years where \(a\) is its distance in astronomical units. A planet has period 125 years. Find \(a\).

Question 9 – Growth factor [2 pts]

A colony triples every 2 hours, so its hourly growth factor is \(3^{1/2}\). Find the multiplier after 6 hours and after 5 hours (leave the second answer in radical form).

Key facts behind this rational exponents test

Key facts

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

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Algebra 2 chapters: Algebra 2 math.

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