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.
What you'll learn
- Find square roots of perfect squares
- Find cube roots of perfect cubes
- Estimate non-perfect square roots between two whole numbers
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.
- Ask: what number times itself equals 64? 8 x 8 = 64, so sqrt(64) = 8.
- Ask: what number cubed equals 27? 3 x 3 x 3 = 27.
- So the cube root of 27 is 3.
Answer: 8 and 3
Practice problems
1. Find sqrt(25).
Show solution
- Warm-up: First identify exactly what the question is asking: Find sqrt(25).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 5 x 5 = 25.
- So sqrt(25) = 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
- Warm-up: First identify exactly what the question is asking: Find sqrt(81).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 9 x 9 = 81.
- So sqrt(81) = 9.
- 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
- Warm-up: First identify exactly what the question is asking: Find the cube root of 8.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 2 x 2 x 2 = 8.
- So the cube root of 8 is 2.
- 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
- Core Practice: First identify exactly what the question is asking: Find the cube root of 125.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 5 x 5 x 5 = 125.
- So the cube root of 125 is 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
- Core Practice: First identify exactly what the question is asking: Find sqrt(100).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 10 x 10 = 100.
- So sqrt(100) = 10.
- 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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