What is a square root?

The square root of x is the number that multiplied by itself gives x. The principal (positive) root is the usual answer: √49 = 7, even though (-7)² is also 49.

Can I take the square root of a negative number?

Not within the real numbers — no real number squared is negative. Mathematicians extend to imaginary numbers, where √-9 = 3i, used in engineering and physics.

How do I simplify a square root?

Factor out the largest perfect square: √48 = √(16 x 3) = 4√3. Exact radical form like 4√3 is often preferred in algebra over a rounded decimal.

How do I estimate a square root by hand?

Bracket the number between perfect squares, then refine: √40 is between 6 (36) and 7 (49), closer to 6.3. Averaging a guess g with x/g (Newton's method) converges very fast.

Why are most square roots irrational?

Unless a whole number is a perfect square, its square root is irrational — the decimal never repeats or ends. That's why √2 ≈ 1.41421356... can only be written exactly in radical form.

Square Root Calculator

Calculate the square root of any number with step-by-step simplification. Supports decimals, fractions, and exact radica

Calculation Type

What do you want to calculate?

Input Values

Number

Decimal Precision

Common Perfect Squares

√1 = 1

√36 = 6

√4 = 2

√49 = 7

√9 = 3

√64 = 8

√16 = 4

√81 = 9

√25 = 5

√100 = 10

Square Root

√16

Exact Result

√16 = 4

Simplified Radical

Factored form

Squared

16.000000

(4.00)²

Reciprocal

1/4.00

0.250000

Radical Simplification

√16 = 4
4 × √1

How it works

A square root answers “what number, multiplied by itself, gives this?” It's the inverse of squaring. Every positive number has two square roots (one positive, one negative); the calculator reports the principal (positive) one.

Square root

√n = x   such that   x² = n

n: the number (radicand)
x: the square root

Worked example

Find √144

Which x gives x² = 144?
12 × 12 = 144

√144 = 12.

Good to know

Perfect squares (1, 4, 9, 16, 25…) have whole-number roots; everything else is irrational and only approximated.
You can't take the real square root of a negative number — that requires imaginary numbers.
√(ab) = √a · √b, which helps simplify roots by pulling out perfect-square factors.