Radical Expressions and Equations
A free College Algebra lesson from the “Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.
Radicals simplify by factoring out perfect squares. Radical equations often require isolating the radical and squaring both sides, then checking solutions.
What you'll learn
- Simplify radicals
- Solve simple radical equations
- Check for extraneous solutions
Why it matters: Distance formulas, free-fall time, standard deviation, and square-root models all use radicals. Simplifying exposes perfect-square structure, while checking after squaring protects against extraneous answers.
Worked example
Problem. Solve sqrt(x + 5) = 6.
- Square both sides: x + 5 = 36.
- Subtract 5.
- x = 31.
Answer: 31
Practice problems
1. Simplify sqrt(50).
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify sqrt(50).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 50 = 25 x 2.
- Check the result by substituting or estimating: the response should match 5sqrt(2) and make sense in the original problem.
Answer: 5sqrt(2)
2. Solve sqrt(x - 4) = 7.
Show solution
- Core Practice: First identify exactly what the question is asking: Solve sqrt(x - 4) = 7.
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- x - 4 = 49.
- Check the result by substituting or estimating: the response should match 53 and make sense in the original problem.
Answer: 53
3. Simplify sqrt(72).
Show solution
- Challenge: First identify exactly what the question is asking: Simplify sqrt(72).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 72 = 36 x 2.
- Check the result by substituting or estimating: the response should match 6sqrt(2) and make sense in the original problem.
Answer: 6sqrt(2)
4. Simplify sqrt(48).
Show solution
- Simplifying Radicals: First identify exactly what the question is asking: Simplify sqrt(48).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 48 = 16 x 3.
- Check the result by substituting or estimating: the response should match 4sqrt(3) and make sense in the original problem.
Answer: 4sqrt(3)
5. Solve sqrt(x) = 9.
Show solution
- Radical Equations: First identify exactly what the question is asking: Solve sqrt(x) = 9.
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- Square both sides.
- Check the result by substituting or estimating: the response should match 81 and make sense in the original problem.
Answer: 81
Practice this interactively with instant feedback and an AI tutor.
Practice Radical Expressions and Equations Take the free placement check