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Radical Expressions

A free Algebra II lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.

Radical expressions involve roots, most often square roots. Simplifying radicals means pulling out perfect-square factors while leaving the non-square part inside the radical. This matters because simplified radical form makes expressions easier to compare, combine, and solve. When practicing, factor the radicand into a perfect square times a leftover factor, then take the square root of the perfect square. A common mistake is splitting addition inside a radical, such as treating sqrt(a + b) like sqrt(a) + sqrt(b). That property does not work.

What you'll learn

Why it matters: Pythagorean distances, free-fall times, standard-deviation calculations, and signal-amplitude math all simplify with radical expressions. Pulling out perfect-square factors — like sqrt(72) = 6 · sqrt(2) — is what makes an answer compact and comparable.

Worked example

Problem. Simplify sqrt(72).

  1. 72 = 36 x 2.
  2. sqrt(36) = 6.
  3. sqrt(72) = 6sqrt(2).

Answer: 6sqrt(2)

Practice problems

1. Simplify sqrt(50).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify sqrt(50).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 50 = 25 x 2.
  4. Check the result by substituting or estimating: the response should match 5sqrt(2) and make sense in the original problem.

Answer: 5sqrt(2)

2. Simplify sqrt(48).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify sqrt(48).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 48 = 16 x 3.
  4. Check the result by substituting or estimating: the response should match 4sqrt(3) and make sense in the original problem.

Answer: 4sqrt(3)

3. Simplify 2sqrt(5) + 3sqrt(5).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify 2sqrt(5) + 3sqrt(5).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. Add coefficients of like radicals.
  4. Check the result by substituting or estimating: the response should match 5sqrt(5) and make sense in the original problem.

Answer: 5sqrt(5)

4. Simplify sqrt(18) + sqrt(8).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify sqrt(18) + sqrt(8).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. sqrt(18)=3sqrt(2) and sqrt(8)=2sqrt(2).
  4. Check the result by substituting or estimating: the response should match 5sqrt(2) and make sense in the original problem.

Answer: 5sqrt(2)

5. Which radical is already simplified?

Choices: sqrt(7) · sqrt(12) · sqrt(20) · sqrt(45)

Show solution
  1. Challenge: First identify exactly what the question is asking: Which radical is already simplified?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 7 has no perfect-square factor other than 1.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sqrt(7)

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