Square Root Calculator

Get the decimal square root and, for whole numbers, the simplified radical form used in algebra and geometry.

Square root of x

Square root of 728.4852813742
Simplified radical form
6√2
Check
8.4852813742² = 72

72 also has a negative square root, −8.4852813742, because (−8.4852813742)² = 72.

What a square root is

The square root of x is the number that, multiplied by itself, gives x. Every positive number has two square roots, one positive and one negative; the symbol √ means the positive (principal) one.

√x = y ⇔ y² = x, y ≥ 0

Simplifying a radical

Simplify √72

  1. Find the largest perfect square that divides 72: 36
  2. √72 = √(36 × 2) = √36 × √2
  3. √36 = 6

√72 = 6√2 ≈ 8.4853

Estimating without a calculator

Find the perfect squares either side: 8² = 64 and 9² = 81, so √72 is between 8 and 9 and a little closer to 8. Linear interpolation gives 8 + 8/17 ≈ 8.47, close to the true 8.4853.

Negative numbers

No real number squared is negative, so √−9 is not real. In complex numbers it is 3i, where i = √−1. The calculator shows this form for negative inputs.