Radical Operations Worksheet
Simplify and combine radicals, then rationalize denominators. Includes a worked example, printable practice, and an answer key with explanations.
The answer key follows the practice below. In the editor, remove the final answer-key section before sharing a student copy.
Use This WorksheetIncludes
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Topics
Radical Operations
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Learning goal and worked example
Simplify and combine radicals, then rationalize denominators.
√50 + √8 = 5√2 + 2√2 = 7√2. Combine like radicals only after simplifying. For cube roots, factors that are perfect cubes come outside.
Prerequisites: factorization and distributive multiplication. Principal square roots are nonnegative; cube roots can be negative.
Read each question carefully and show all your work. Simplify all radical expressions completely.
1. Simplify the following radical expression: √72
2. Simplify: ³√108
3. The sum of 5√3 and 2√3 is .
4. Subtract: 7√5 - 3√20 = .
5. What is the product of (√6)(√10)?
√30
2√15
4√15
√16
6. Multiply: (2√3 + 1)(√3 - 4)
6 - 7√3
2 + 7√3
2 - 7√3
6 - 9√3
7. The expression 1/√2 can be rationalized to √2/2.
True
False
8. (√18)/(√2) simplifies to 3.
True
False
9. Rationalize the denominator: (3 + √2) / (√2 - 5)
10. For real x and y, simplify: ∛(16x⁵y³) · ∛(4x²y⁴). Each pair of parentheses is entirely inside its cube root.
Answer key and worked solutions
1. √72=√(36×2)=6√2. 2. ∛108=∛(27×4)=3∛4.
3. 7√3. 4. 7√5−3(2√5)=√5.
5. b: √60=2√15. 6. c: 6−8√3+√3−4=2−7√3.
7. True: multiply numerator and denominator by √2. 8. True: √18/√2=√9=3.
9. Multiply by (√2+5)/(√2+5): numerator = 17+8√2, denominator = 2−25=−23. Answer: −(17+8√2)/23.
10. ∛(64x⁷y⁷)=4x²y²∛(xy). Extract 64, x⁶, and y⁶ as perfect cubes. This remains valid for negative real x or y.
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