Square Root Calculator
What a square root is
The square root of a number x is the value that, multiplied by itself, gives x: √49 = 7 because 7 × 7 = 49. Every positive number has two square roots (7 and −7); the symbol √ refers to the positive one.
Perfect squares to remember
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Knowing them makes estimates easy: √50 is a little more than 7 (since 7² = 49), and the calculator gives 7.07.
Irrational roots and simplification
Most square roots are irrational: their decimals never end or repeat. √2 = 1.41421356… is the diagonal of a square of side 1. Roots can often be simplified by extracting a perfect square: √50 = √(25 × 2) = 5√2 ≈ 5 × 1.414 = 7.07.
Calculating by hand: the Babylonian method
Start with a guess, then repeatedly average it with x divided by the guess:
- Guess for √50: 7
- (7 + 50 ÷ 7) ÷ 2 = (7 + 7.1429) ÷ 2 = 7.0714
- One more step gives 7.07107, already correct to five decimals.
This ancient method is still close to how computers calculate roots.
Everyday uses
- Pythagoras' theorem: the diagonal of a rectangle is √(length² + width²). A 3 m × 4 m room has a 5 m diagonal; a TV screen 121 cm wide and 68 cm high has a diagonal of √(121² + 68²) ≈ 139 cm, i.e. 55 inches.
- Side of a square from its area: a 30 m² square room measures √30 ≈ 5.48 m per side.
- Statistics: the standard deviation is the square root of the variance.
Negative numbers have no real square root (√−4 is an imaginary number, 2i), so the calculator shows an error for them. For cube roots or other powers, use the power calculator with a fractional exponent: x^(1/3).
Frequently Asked Questions
What is the square root of 2?
About 1.41421. It is an irrational number: its decimals never end or repeat.
Can you take the square root of a negative number?
Not among real numbers. In mathematics, √−1 is defined as the imaginary unit i, so √−4 = 2i.
How do I calculate a cube root?
Use the power calculator with an exponent of 1/3 (0.3333…). For example, 27^(1/3) = 3.