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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

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response). Counts include repeated prompts and variants.

Practice problems

1. Find sqrt(25).

Show solution
  1. 5 x 5 = 25.
  2. So sqrt(25) = 5.

Answer: 5

2. Find sqrt(81).

Show solution
  1. 9 x 9 = 81.
  2. So sqrt(81) = 9.

Answer: 9

3. Find the cube root of 8.

Show solution
  1. 2 x 2 x 2 = 8.
  2. So the cube root of 8 is 2.

Answer: 2

4. Find the cube root of 125.

Show solution
  1. 5 x 5 x 5 = 125.
  2. So the cube root of 125 is 5.

Answer: 5

5. Find sqrt(100).

Show solution
  1. 10 x 10 = 100.
  2. So sqrt(100) = 10.

Answer: 10

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