Calculators

Square Root Calculator

Square, cube and higher roots, with the simplest radical form such as √72 = 6√2.

How it works

The nth root of x is the number that gives x when multiplied by itself n times. To simplify a radical, the calculator factors out the largest perfect nth power that divides the number and moves its root outside the sign. Negative numbers have real odd roots (∛−27 = −3) but no real even roots.

ⁿ√x = x^(1/n) | √(a²·b) = a√b
√
Square root (n = 2)
∛
Cube root (n = 3)
Perfect square
A whole number whose square root is whole, like 49
Radicand
The number under the root sign

Worked example

The square root of 72.

  1. 1The largest perfect square dividing 72 is 36
  2. 2√72 = √(36 × 2) = √36 × √2
  3. 3= 6√2
  4. 4As a decimal: 6 × 1.41421356 = 8.48528137

√72 = 6√2 ≈ 8.4853. 72 is not a perfect square; it lies between 64 (8²) and 81 (9²).

Frequently asked questions

How do I find a square root without a calculator?

Find the perfect squares either side, then refine. √50 lies between 7 (√49) and 8 (√64), close to 7. A good next guess is to average your guess with the number divided by it: (7 + 50 ÷ 7) ÷ 2 = 7.071, already correct to three decimals.

Can you take the square root of a negative number?

Not as a real number, because any real number squared is positive or zero. In complex numbers √−1 is called i, so √−9 = 3i. Odd roots of negatives are fine: ∛−8 = −2.

Why does √ give only one answer when 3² and (−3)² are both 9?

Both 3 and −3 are square roots of 9, but the √ symbol means the principal, non-negative root, so √9 = 3. When solving x² = 9 you write x = ±3 to include both.

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