Softonic

Square Root Calculator

Step 1 / 4 Square root

Enter any number. For non-negative inputs, the calculator returns the principal square root as a decimal; for positive whole numbers it also shows a simplified radical where possible: √72 = 6√2 and √200 = 10√2. Perfect squares return whole numbers. Negative inputs are flagged as complex, and the guide below explains how √(−x) becomes i√x.

How the root is computed

  1. 1

    Enter the radicand

    Type the number under the radical sign. It can be positive, negative or zero.

  2. 2

    Read the decimal result

    For non-negative inputs, the calculator returns the principal square root using standard floating-point square-root arithmetic.

  3. 3

    Check the simplified radical

    For positive integers, perfect-square factors are pulled outside the radical. √72 = √(36 × 2) = 6√2.

  4. 4

    Use the notes to verify it

    The examples below show the factor method, common decimal values and the difference between real and complex roots.

Perfect squares to know

n n² √(n²)
1 1 1
2 4 2
3 9 3
4 16 4
5 25 5
10 100 10
11 121 11
12 144 12
13 169 13
14 196 14
15 225 15
16 256 16
25 625 25

Simplifying non-perfect squares

For whole numbers, look for the largest perfect-square factor:

  • √50 = √(25 × 2) = 5√2
  • √72 = √(36 × 2) = 6√2
  • √108 = √(36 × 3) = 6√3
  • √500 = √(100 × 5) = 10√5
  • √1000 = √(100 × 10) = 10√10

If a non-square factor remains, keep it under the radical. For example, √180 = √(36 × 5) = 6√5, not √(4 × 45) = 2√45, because 45 still contains the perfect-square factor 9.

Common decimal values and uses

  • √2 ≈ 1.41421 (classroom geometry: the diagonal of a 1 × 1 square)
  • √3 ≈ 1.73205 (space diagonal of a unit cube)
  • √5 ≈ 2.23607 (appears in the golden ratio (1+√5)/2)
  • √7 ≈ 2.64575
  • √π ≈ 1.77245 (statistics and Gaussian integrals)
  • √10 ≈ 3.16228
  • √(1000) ≈ 31.6228 (a 10x increase multiplies the square root by about 3.16)

In construction or measurement work, square roots often turn an area into a side length: a square patio of 49 m² has side length √49 = 7 m, while an 80 m² square space has side length √80 = 4√5 m ≈ 8.94 m.

Negative numbers and complex roots

The square root of a negative number is not a real number. In the complex numbers, √(−x) = i√x for positive x, so √(−4) = 2i. The live calculator flags negative inputs as complex rather than showing a real decimal.

Square root vs nth root

This calculator handles square (2nd) roots. For cube roots, fourth roots and other indices, use a general nth-root tool. Key identities:

  • √(ab) = √a × √b (only if a and b are non-negative)
  • √(a/b) = √a / √b (only if b > 0)
  • (√a)² = a (only if a >= 0)

History pointer

The radical symbol √ evolved from the letter r (for radix, root in Latin) in the 1500s. The horizontal bar, called the vinculum, was added in the 17th century to show exactly what belongs under the root.

Frequently Asked Questions

Every positive number has two square roots: +r and −r. The radical symbol √ returns the principal root, the non-negative one. When solving quadratic equations, you usually consider both roots.

Just 5 by convention. √ returns the principal root. When solving x² = 25, both 5 and −5 satisfy the equation, so you write x = ±5.

Common methods include factoring out perfect squares, the digit-by-digit long-division algorithm and Newton’s method: x_new = (x + a/x)/2. Newton’s method converges quickly; a few iterations are usually enough for classroom checks.

A classic proof uses contradiction. If √2 = p/q in lowest terms, then 2q² = p², so p is even. Write p = 2k, which gives q² = 2k², so q is even too. That contradicts the assumption that p/q was in lowest terms, so √2 cannot be a fraction.

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