Radicals and Rational Exponents: Algebra 2 Math Test – PDF Download

Question 1: Roots and powers (2 points)

Evaluate each expression without a calculator.

1. \(\sqrt[3]{-125}\) (1 point)
2. \(81^{3/4}\) (1 point)

Question 2: Radical and exponent forms (2 points)

1. Write \(x^{5/3}\) as a radical. (1 point)
2. Write \(\sqrt[4]{y^{3}}\) with a rational exponent. (1 point)

Question 3: Simplify radicals (2 points)

Simplify each answer.

1. Find the exact length \(c\) of the hypotenuse of the triangle shown. (1 point)
2. Simplify \(\sqrt[3]{40x^{4}}\), where \(x\) is positive. (1 point)

Question 4: Combine radicals (2 points)

1. Simplify \(5\sqrt{12}-\sqrt{48}+\sqrt{75}\). (1 point)
2. Rationalize the denominator of \(\frac{12}{\sqrt{8}}\). (1 point)

Question 5: Exponent properties (2 points)

Simplify. Assume \(x\) and \(a\) are positive.

1. \(x^{2/3}\cdot x^{3/4}\) (1 point)
2. \((16a^{4})^{3/4}\) (1 point)

Question 6: Solve equations (3 points)

Solve each equation.

1. \(2\sqrt{3x-2}-4=6\) (1 point)
2. \(x^{3/2}=64\) (1 point)
3. \(\sqrt[3]{x+5}=-2\) (1 point)

Question 7: Extraneous solution (3 points)

The graphs of \(y=\sqrt{x+1}\) and \(y=x-1\) are shown.

1. Use the graph to estimate the solution of \(\sqrt{x+1}=x-1\). (1 point)
2. Solve the equation algebraically. (1 point)
3. Check each candidate and name any extraneous solution. (1 point)

Question 8: Cube-shaped box (4 points)

A cube-shaped gift box has a volume of \(192\) cubic inches.

1. Find the exact edge length in simplest radical form. (1 point)
2. Round the edge length to the nearest hundredth. (1 point)
3. Find the exact total surface area. (1 point)
4. The box must fit a shelf \(6\) inches high. Does it fit? Explain. (1 point)

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