Undoing powers
Roots undo powers:
| Power | Root | Example |
|---|---|---|
| Squaring | Square root (√) | 8² = 64 → √64 = 8 |
| Cubing | Cube root (∛) | 4³ = 64 → ∛64 = 4 |
The symbol √ is called the radical sign.
Square roots
Ask: what number times itself gives this?
- √49 = 7 because 7 × 7 = 49
- √100 = 10
- √0.25 = 0.5
Both 7 and −7 square to 49, but √49 means the positive one, called the principal square root. If we want both, we write ±√49 = ±7.
You can’t take the square root of a negative number using real numbers, because no real number squared is negative.
Cube roots
Ask: what number used three times gives this?
- ∛27 = 3
- ∛1,000 = 10
- ∛(−8) = −2 (because (−2)³ = −8)
Cube roots of negative numbers work fine.
Estimating roots
If the number isn’t a perfect square, estimate using the perfect squares on either side.
Example: √50
- 7² = 49 and 8² = 64.
- 50 is just above 49, so √50 is just above 7: about 7.07.
Example: √30
- 5² = 25, 6² = 36.
- 30 is about halfway, a bit closer to 25, so √30 ≈ 5.5 (actual 5.477).
Roots of large and small numbers
- √3,600 = 60 (since √36 = 6, and √100 = 10)
- √0.09 = 0.3
- ∛8,000 = 20
Roots in geometry
- Side of a square from its area: area 64 cm² → side √64 = 8 cm.
- Edge of a cube from its volume: volume 125 cm³ → edge ∛125 = 5 cm.
Worked examples
Example 1: √64 = ?
8 × 8 = 64 → 8.
Example 2: ∛216 = ?
6 × 6 × 6 = 216 → 6.
Example 3: Between which two whole numbers is √90?
9² = 81, 10² = 100 → between 9 and 10 (closer to 9.5).
Common mistakes
- Dividing by 2 instead of finding the root: √64 is 8, not 32.
- Dividing by 3 for a cube root: ∛27 is 3, not 9.
- Forgetting negative cube roots exist.
Try it yourself
- √121
- ∛64
- ∛(−125)
- Estimate √20 to one decimal place.
- A square rug has area 2.25 m². How long is a side?
Answers: 1) 11 2) 4 3) −5 4) about 4.5 (actual 4.47) 5) 1.5 m
Quick recap
- The square root undoes squaring; the cube root undoes cubing.
- √ gives the principal square root (positive).
- Estimate roots between nearby perfect powers.