What is a square root?
Squaring means multiplying a number by itself: 6² = 6 × 6 = 36. The square root undoes this: √36 = 6.
The sign √ is called the radical sign. The number under it is the radicand.
Both 6 × 6 and (−6) × (−6) give 36. So 36 has two square roots, +6 and −6. The symbol √36 means only the positive one (the principal root). To solve x² = 36 we write x = ±6.
A negative number has no real square root, because any real number times itself is never negative.
Perfect squares and how to spot them
Perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, … Learn them up to 15² = 225.
- A perfect square never ends in 2, 3, 7 or 8.
- In its prime factorisation every prime appears an even number of times. 144 = 2⁴ × 3², so √144 = 2² × 3 = 12.
Square roots of fractions and decimals
√(9/16) = 3/4. √0.49 = 0.7 because 0.7 × 0.7 = 0.49.
Estimating the root of a non-perfect square
Find the perfect squares on each side. For √40: 36 < 40 < 49, so 6 < √40 < 7. 40 is closer to 36, so the root is nearer 6: about 6.3.
Check by squaring: 6.3² = 39.69, 6.4² = 40.96. So √40 ≈ 6.32.
Roots like √2, √3, √20 are irrational: their decimals never end or repeat. Also, √(x²) = |x|.
Rules for roots and simplifying (surds)
- √(a × b) = √a × √b (a, b ≥ 0)
- √(a ÷ b) = √a ÷ √b (b > 0)
- But √(a + b) is not √a + √b. √(9 + 16) = 5, not 7.
To simplify, pull out the largest perfect square factor: √72 = √36 × √2 = 6√2.
Like roots add like like terms: 3√2 + 5√2 = 8√2. Unlike roots cannot be joined: √2 + √3 stays as it is.
Cube roots
The cube root ∛n is the number used three times: ∛64 = 4 since 4 × 4 × 4 = 64. A negative number has a real cube root: ∛(−8) = −2.
Key formulas and definitions
- √n = s means s × s = n and s ≥ 0
- x² = n → x = ±√n
- √(ab) = √a · √b
- √(a/b) = √a / √b
- √(a + b) ≠ √a + √b
- √(x²) = |x|
- ∛(n) = c means c³ = n
Worked examples
1. Find √196.
196 = 2² × 7². √196 = 2 × 7 = 14. Check: 14 × 14 = 196.
2. Between which two whole numbers is √75?
64 < 75 < 81, so 8 < √75 < 9.
3. Estimate √30 to 1 decimal place.
25 < 30 < 36 → between 5 and 6. 5.4² = 29.16, 5.5² = 30.25. 30 is closer to 30.25, so √30 ≈ 5.5.
4. Simplify √48.
48 = 16 × 3. √48 = √16 × √3 = 4√3.
5. Simplify 2√18 + √50.
2√18 = 2 × 3√2 = 6√2. √50 = 5√2. Total = 11√2.
6. A square floor has area 12.25 m². How long is one side?
Side = √12.25 = 3.5 m, since 3.5 × 3.5 = 12.25.
Common mistakes
- Writing √(9 + 16) = 3 + 4 = 7. Add first: √25 = 5.
- Forgetting the ± when solving x² = 49: x = 7 or x = −7.
- Thinking √(−9) = −3. (−3)² = 9, not −9; there is no real answer.
- Stopping too early when simplifying: √72 = 2√18 is not finished; it is 6√2.