Square root simplifier

Last updated 2026-10-02

This square root simplifier writes any square root in simplest radical form and shows how: the prime factorization, the pairs, and the largest perfect square factor. Enter 72 and you get $6\sqrt{2}$ , about 8.4853.

This calculator needs JavaScript. Without it, simplifying square roots walks through the method with examples.

How to simplify a square root

Find the largest perfect square that divides the number. The perfect squares are 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. For 72 the largest one is 36.
Split the root. $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$ .
Take the square root of the perfect square. $\sqrt{36} = 6$ , so $\sqrt{72} = 6\sqrt{2}$ .
Check it. Square the answer: $(6\sqrt{2})^2 = 36 \times 2 = 72$ .

If you cannot spot the perfect square, factor the number into primes and pair them up: $72 = 2 \times 2 \times 2 \times 3 \times 3$ . Each pair comes out of the root as one number, here a 2 and a 3, and what is left stays inside: $2 \times 3 \times \sqrt{2} = 6\sqrt{2}$ .

Two examples

Example 1: the square root of 75

The largest perfect square factor of 75 is 25, since $75 = 25 \times 3$ .
$\sqrt{75} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$ .

So $\sqrt{75} = 5\sqrt{3}$ , about 8.6603. Check: $25 \times 3 = 75$ .

Example 2: the square root of 98

The prime factors are $98 = 2 \times 7 \times 7$ . The 7s pair up.
One 7 comes out and the 2 stays in: $\sqrt{98} = 7\sqrt{2}$ .

So $\sqrt{98} = 7\sqrt{2}$ , about 9.8995. Check: $49 \times 2 = 98$ .

Simplified square roots to know

Root	Simplest form	Decimal
$\sqrt{8}$	$2\sqrt{2}$	2.8284
$\sqrt{12}$	$2\sqrt{3}$	3.4641
$\sqrt{18}$	$3\sqrt{2}$	4.2426
$\sqrt{20}$	$2\sqrt{5}$	4.4721
$\sqrt{48}$	$4\sqrt{3}$	6.9282
$\sqrt{50}$	$5\sqrt{2}$	7.0711

The perfect squares chart lists the squares to look for, and simplifying square roots covers the method in depth.

Questions

What is simplest radical form?

A square root is in simplest radical form when the number under the root has no perfect square factor other than 1, there is no fraction under the root, and there is no root in a denominator. $\sqrt{72}$ is not simplest because 36 divides 72. $6\sqrt{2}$ is.

What if the number is a perfect square?

Then the root is a whole number. The square root of 144 is exactly 12, and the simplifier says so.

What about a negative number?

A negative number has no real square root. The simplifier gives the imaginary answer: $\sqrt{-50} = 5i\sqrt{2}$ , where $i = \sqrt{-1}$ .

When the root is one step in a longer problem, like the quadratic formula or the Pythagorean theorem, photograph the problem in AI Math Tutor and check each step.