How to Calculate Roots
A root is the inverse of an exponent. The square root of a number asks: what value multiplied by itself gives this number? This calculator shows both exact and decimal results for square roots, cube roots, and nth roots.
Square Root (√)
Find what number × itself = your number.
- √4 = 2 (because 2 × 2 = 4)
- √9 = 3 (because 3 × 3 = 9)
- √2 ≈ 1.41421 (irrational — decimal never terminates or repeats)
Cube Root (∛)
Find what number × itself × itself = your number.
- ∛8 = 2 (because 2 × 2 × 2 = 8)
- ∛27 = 3
- ∛-8 = -2 (cube roots can be negative, unlike square roots)
Perfect Squares (1-400)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Recognizing perfect squares makes mental math and radical simplification much faster.
How to Simplify Radicals
To simplify a square root, factor out the largest perfect square. The product rule states: √(a × b) = √a × √b.
Example: √72
- Find the largest perfect square factor: 72 = 36 × 2
- √72 = √(36 × 2) = √36 × √2
- = 6√2 ≈ 8.485
More Examples
- √50 = √(25 × 2) = 5√2
- √200 = √(100 × 2) = 10√2
- √48 = √(16 × 3) = 4√3
- √75 = √(25 × 3) = 5√3
- √98 = √(49 × 2) = 7√2
Square Roots in Real Life
- Distance formula: The distance between two points uses √((x₂-x₁)² + (y₂-y₁)²)
- Pythagorean theorem: Find the hypotenuse with c = √(a² + b²). A 3×4 rectangle has a diagonal of √(9+16) = √25 = 5
- Standard deviation: Statistical spread is calculated as the square root of variance
- Screen size: TV diagonal = √(width² + height²). A 16:9 TV that's 44" wide × 24.75" tall has a diagonal of √(1936 + 612.56) = √2548.56 ≈ 50.5"
Properties of Square Roots
- √(a × b) = √a × √b — product rule
- √(a ÷ b) = √a ÷ √b — quotient rule
- (√a)² = a — squaring undoes the root
- Negative numbers have no real square root. √(-1) = i (imaginary unit)
- Every positive number has two square roots: positive and negative. √25 = ±5, though the symbol √ conventionally means the positive root