Square Root Calculator

What a square root is

The square root of a number $x$ is the value that, multiplied by itself, produces $x$:

$\sqrt{x} = y \quad \text{means} \quad y \times y = x$

Since $6 \times 6 = 36$, the square root of $36$ is $6$ — written $\sqrt{36} = 6$. A square root undoes squaring, the same way division undoes multiplication.

The principal root: why √36 = 6, not −6

Both $6$ and $-6$ square to $36$, since $(-6) \times (-6) = 36$ too. But the radical symbol is defined to always return the non-negative root, called the principal square root. So $\sqrt{36} = 6$ by convention — if a problem needs both roots, it is written explicitly as $\pm\sqrt{36} = \pm 6$. This calculator, like every square root symbol, always shows the principal (non-negative) root for a positive input.

Perfect squares vs. simplified radicals

A perfect square is a whole number that is some integer squared — $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots$ Its square root is an exact whole number:

Most numbers are not perfect squares, so their square root is irrational — a never-ending, non-repeating decimal. Rather than truncate that decimal, the exact answer is written as a simplified radical: a whole-number coefficient times the smallest possible root. For example, $\sqrt{44} = 2\sqrt{11}$, because $44 = 4 \times 11$ and $\sqrt{4} = 2$ can be pulled out from under the root, leaving $11$ — which has no square factors left to extract.

Simplifying a radical, step by step

To simplify $\sqrt{72}$ by hand:

1. Factor into primes: $72 = 2^3 \times 3^2$.
2. Group into pairs: the $3^2$ is a complete pair; the $2^3$ has one complete pair of 2s ($2^2$) with one $2$ left over.
3. Pull each pair out as its square root: $2^2 \to 2$ and $3^2 \to 3$, multiplied together as the coefficient: $2 \times 3 = 6$.
4. Leave the leftover factor under the root: the unpaired $2$ stays inside, giving $\sqrt{72} = 6\sqrt{2}$.

This calculator shows that same prime-factorization breakdown for whatever number you enter.

Negative numbers and the imaginary unit i

No real number squares to a negative value, since a negative times a negative is always positive. So the square root of a negative number isn't a real number — it's imaginary. The imaginary unit $i$ is defined as $i = \sqrt{-1}$, which lets any negative square root be written exactly:

$\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i$

The same simplification rules apply to the positive part before the $i$ is attached — for example $\sqrt{-8} = 2\sqrt{2}\,i$, since $\sqrt{8}$ simplifies to $2\sqrt{2}$ first.

The inverse: squaring

Taking a square root and squaring undo each other: $(\sqrt{x})^2 = x$ for $x \geq 0$. If $\sqrt{y} = x$, then $x^2 = y$ — so $\sqrt{49} = 7$ and $7^2 = 49$ are the same fact, read in opposite directions. Use the Square calculator to compute $x^2$ directly.

Worked examples

Example 1 — perfect square: $\sqrt{9} = 3$, exactly, since $3 \times 3 = 9$.

Example 2 — simplified radical: $\sqrt{44} = 2\sqrt{11} \approx 6.63325$, since $44 = 4 \times 11$.

Example 3 — negative input: $\sqrt{-9} = 3i$, an imaginary result, since no real number squares to $-9$.

Example 4 — already simplified: $\sqrt{2} \approx 1.414214$ stays as $\sqrt{2}$ — 2 has no square factors to pull out.

Where square roots show up

Square roots appear throughout geometry (the Pythagorean theorem, $c = \sqrt{a^2 + b^2}$), physics (falling-object time, RMS voltage), statistics (standard deviation is the square root of variance), and finance (compound annual growth rate). Squaring — the exact inverse — is common enough to get its own Square calculator, and roots beyond index 2 are covered by the Cube Root and general Nth Root calculators.