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Square Roots and Cube Roots

A free Pre-Algebra lesson from the “Decimals, Roots, and the Real Number System” unit, with a worked example and practice problems including step-by-step solutions.

A square root of n is a number that multiplied by itself gives n. A cube root of n is a number that multiplied by itself three times gives n. Perfect squares (1, 4, 9, 16, 25, ...) and perfect cubes (1, 8, 27, 64, ...) have whole-number roots. Non-perfect roots fall between two whole numbers. In Decimals, Roots, and the Real Number System, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

What you'll learn

Why it matters: Finding the side length of a square given its area (square root) and the edge length of a cube given its volume (cube root) are everyday applications, along with reading distance formulas and physics equations.

Worked example

Problem. Find sqrt(64) and the cube root of 27.

  1. Ask: what number times itself equals 64? 8 x 8 = 64, so sqrt(64) = 8.
  2. Ask: what number cubed equals 27? 3 x 3 x 3 = 27.
  3. So the cube root of 27 is 3.
  4. Connect the calculation back to Square Roots and Cube Roots so the method, not just the arithmetic, is clear.

Answer: 8 and 3

Practice problems

1. Find sqrt(25).

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

Answer: 5

2. Find sqrt(81).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find sqrt(81).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 9 x 9 = 81.
  4. So sqrt(81) = 9.
  5. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

3. Find the cube root of 8.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the cube root of 8.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 2 x 2 x 2 = 8.
  4. So the cube root of 8 is 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

4. Find the cube root of 125.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the cube root of 125.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 5 x 5 x 5 = 125.
  4. So the cube root of 125 is 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. Find sqrt(100).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find sqrt(100).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 10 x 10 = 100.
  4. So sqrt(100) = 10.
  5. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

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