Rational Exponents Worksheet with Answers – Algebra 2 (Free PDF)

Problem 1 Evaluating roots ★★★

Evaluate each root.

1. ∛(−125)
2. ⁴√(81)
3. √(4964)
4. ⁵√(32)

Problem 2 Evaluating rational exponents ★★★

Evaluate each expression without a calculator.

1. 25^1/2
2. 8^2/3
3. 16^3/4
4. 49^3/2

Problem 3 Radical form to exponent form ★★★

Rewrite each expression using a rational exponent.

1. √x
2. ∛(x²)
3. ⁴√(x³)
4. (⁵√y)²

Problem 4 Exponent form to radical form ★★★

Rewrite each expression as a radical.

1. x^1/3
2. y^5/2
3. 7^3/4

Problem 5 Simplifying numeric radicals ★★★

Simplify each radical.

1. √162
2. ∛40
3. √(1,176)

Problem 6 Reading a root graph ★★★

The graph of y = x^1/2 is shown. Use it to find 9^1/2 and 4^1/2. Then state two consecutive integers between which 12^1/2 lies, and check with a calculation.

Problem 7 Properties of rational exponents ★★★

Simplify each expression. Assume x > 0.

1. x^1/2 · x^1/3
2. (x^2/3)^3/4
3. x^3/4x^1/4

Problem 8 Radicals with variables ★★★

Simplify. Assume all variables are positive.

1. √(18x⁵)
2. ∛(54a⁴)

Problem 9 Combining like radicals ★★★

Simplify 5√12 − 2√27 + √75.

Problem 10 Multiplying radical expressions ★★★

Multiply and simplify.

1. (3 + √5)(3 − √5)
2. (2√3 + 1)²

Problem 11 Rationalizing denominators ★★★

Write each expression with a rational denominator.

1. 6√8
2. 5√7 − 2

Problem 12 Solving with a graph ★★★

Solve √(x + 6) = x algebraically. Use the figure to explain which candidate is extraneous.

Problem 13 Single radical equations ★★★

Solve each equation and check.

1. √(3x − 5) = 4
2. ∛(2x + 1) = −3

Problem 14 An extraneous solution ★★★

Solve √(2x + 9) = x + 3. Check every answer.

Problem 15 Isolating a radical first ★★★

Solve √(x + 14) − x = 2.

Problem 16 Two radicals ★★★

Solve √(x + 5) + √x = 5.

Problem 17 A rational exponent equation ★★★

Solve (x − 4)^2/3 = 9. Check each answer in the original equation.

Problem 18 Skid marks and speed ★★★

An investigator estimates a car’s speed s (mph) from the length d (feet) of its skid with s = √(21d). Find the skid length for a speed of 42 mph and the exact speed, in simplest radical form, for a 147-foot skid.

Problem 19 Distance to the horizon ★★★

From a lookout that is h feet above the ground, the distance to the horizon is d = √(1.5h) miles. A ranger on a fire tower sees the horizon 12 miles away. How tall is the tower? How far is the horizon for a hiker whose eyes are 6 feet above the ground?

Problem 20 Side of a cube ★★★

A shipping crate shaped like a cube holds 2,744 cubic inches. Use a cube root, written as a rational exponent, to find the side length, and then find the surface area of the crate.