Roots undo powers
Because 12² = 144, √144 = 12. Because 5³ = 125, ∛125 = 5. For positive numbers, squaring and taking the principal square root undo each other; cubing and taking the cube root do the same.
Square Roots and Cube Roots
Think first: If a cube has volume 729 cm³, can you find its edge length without guessing every possible value?
Open practice worksheet
Know
- perfect squares and perfect cubes
- square-root and cube-root notation
- exact and approximate root values
Understand
- roots undo powers
- why √(ab) = √a × √b for positive whole numbers
- why a non-perfect root can be located between consecutive whole numbers
Can do
- evaluate perfect square and cube roots
- estimate non-perfect square and cube roots, then check with a calculator
- apply roots and powers in mixed expressions
Perfect and non-perfect roots
144 is a perfect square. The number 150 is not, but 12² = 144 < 150 < 169 = 13², so √150 lies between 12 and 13.
Exact and approximate roots
A root is exact when its stated value equals the root, such as √144 = 12. For a non-perfect power, a decimal from a calculator is usually approximate: √150 ≈ 12.25 and ∛200 ≈ 5.85. First bound the value using neighbouring perfect powers, then use a calculator to check the estimate.
Root products and mixed expressions
For positive whole numbers, numerical examples verify √(ab) = √a × √b. For example, √(9 × 16) = √144 = 12 and √9 × √16 = 3 × 4 = 12.
Apply brackets, powers and roots before multiplication, division, addition and subtraction:
√81 + ∛27 × 2² = 9 + 3 × 4 = 21
Worked examples
Worked example 1: Evaluate √225
15² = 225, so √225 = 15. Check: 15 × 15 = 225.
Worked example 2: Evaluate ∛1,000
10³ = 1,000, so ∛1,000 = 10.
Worked example 3: Locate √70 between integers
8² = 64 and 9² = 81, so 8 < √70 < 9. It is closer to 8 because 70 is closer to 64.
Worked example 4: Estimate ∛200, then check
5³ = 125 and 6³ = 216, so 5 < ∛200 < 6 and it is closer to 6. A calculator check gives ∛200 ≈ 5.85.
Worked example 5: Evaluate √(25 × 4) in two ways
Directly, √100 = 10. Using the root-product relationship, √25 × √4 = 5 × 2 = 10.
Check your understanding
Reason and apply
1. Evaluate √324 and verify.
Model response: √324 = 18 because 18² = 324.
2. Evaluate ∛512 and verify.
Model response: ∛512 = 8 because 8³ = 512.
3. Locate √115 between consecutive integers and state which it is closer to.
Model response: 10² = 100 and 11² = 121, so 10 < √115 < 11. It is closer to 11 because 115 is 6 below 121 but 15 above 100.
4. A square has area 288 cm². Between which whole numbers is its side length?
Model response: 16² = 256 and 17² = 289, so the side length √288 is between 16 cm and 17 cm, very close to 17 cm.
5. Estimate ∛90 between consecutive whole numbers, state which integer it is closer to, then check with a calculator.
Model response: 4³ = 64 and 5³ = 125, so 4 < ∛90 < 5. It is closer to 4 because 90 is 26 above 64 and 35 below 125. A calculator gives ∛90 ≈ 4.48.
6. Classify each answer as exact or approximate: √196 = 14 and √70 ≈ 8.37. Explain.
Model response: √196 = 14 is exact because 14² = 196. √70 ≈ 8.37 is approximate because 70 is not a perfect square and the decimal has been rounded.
7. Verify √(4 × 49) = √4 × √49, then evaluate √64 + ∛125 × 3².
Model response: Both sides of the root product equal 14: √196 = 14 and 2 × 7 = 14. For the expression, roots and powers first: 8 + 5 × 9 = 8 + 45 = 53.