How to Simplify a Square Root Into Exact Form

The answer key says 6√2 and your calculator says 8.4852813742. You’re fairly sure those are the same thing, but the homework wants exact form, and you can’t get from √72 to 6√2 reliably enough to trust yourself in an exam where the calculator is a decimal-only one.

Simplifying radicals is one of those topics that’s genuinely easy once the method clicks, and genuinely frustrating until it does — mostly because the standard explanation skips the step where you decide which factor to pull out.

What simplifying a radical actually means

A square root is in simplified exact form when the number under the radical has no perfect-square factors left. √72 isn’t simplified because 72 = 36 × 2, and 36 is a perfect square. √2 is simplified because 2 has no square factors at all.

The method is one line: find the largest perfect square that divides the number, take its root out front, and leave the rest inside.

For √72: the perfect squares are 4, 9, 16, 25, 36, 49, 64. The largest one dividing 72 is 36. So √72 = √(36 × 2) = √36 × √2 = 6√2.

If you can’t spot the largest one, take any perfect square factor and repeat. √72 = √(4 × 18) = 2√18, and √18 = √(9 × 2) = 3√2, so 2 × 3√2 = 6√2. Same answer, one extra step.

Why people get stuck here

• Spotting the largest square factor. Most people find 4 immediately and 36 never, which means extra rounds and more chances to slip.
• Stopping too early. 2√18 is a correct factoring but not fully simplified, and it usually loses marks.
• Variables under the radical. √(x⁵) needs the same logic applied to exponents, and the odd leftover power trips people up.
• Fractions in the radicand. Rationalising the denominator is a separate skill bolted onto the same problem.
• Trusting a decimal. 8.485 doesn’t tell you whether the exact answer is 6√2 or 2√18 or something else entirely.

What a good simplification looks like

The radicand has no square factors

Check the final answer: can anything inside the radical still be divided by 4, 9, 16, 25, or 36? If yes, you’re not finished.

The coefficient is fully extracted

Every factor you pulled out multiplies together into a single number out front. 2 × 3√2 should be written as 6√2, not left as a product.

It matches numerically

A quick decimal check catches sign and arithmetic errors. 6√2 ≈ 8.485 and √72 ≈ 8.485, so the transformation held.

Common mistakes to avoid

• Adding radicals that aren’t like terms. √2 + √3 does not become √5. Only matching radicands combine: 3√2 + 5√2 = 8√2.
• Splitting a sum under the radical. √(9 + 16) is 5, not 3 + 4. The product rule works for multiplication, not addition.
• Leaving a radical in the denominator. Most courses want 1/√2 rewritten as √2/2.
• Forgetting the sign. √(x²) is |x|, not x, when x could be negative — this matters more than it seems in later topics.
• Using the decimal as the answer. If the question says “exact form,” a decimal isn’t an answer no matter how many places you give.

How to do it with Square Root Simplifier

Online Tool Store’s Square Root Simplifier reduces radicals to exact form in your browser, which makes it useful for checking work rather than replacing it.

1. Enter the number under the radical — 72, 180, whatever you’re working with.
2. Read the simplified exact form it produces.
3. Before accepting it, try to spot the perfect square factor yourself. That’s the skill the exam is testing.
4. Verify by squaring your coefficient and multiplying by the remaining radicand — 6² × 2 = 72 confirms 6√2.
5. Work through a run of similar numbers to build the habit of recognising 4, 9, 16, 25, 36, 49, and 64 as factors quickly.

For neighbouring topics, the Prime Factorization Tool makes the square-factor hunt much easier, and the Fraction Simplifier handles the rational side.

Frequently asked questions

How do I find the largest perfect square factor quickly?

Prime factorise the number and pair up the primes. 72 = 2³ × 3² gives you a pair of 2s and a pair of 3s, so 2 × 3 = 6 comes out and one leftover 2 stays in — that’s 6√2 without any guessing.

What if the number is prime?

Then it’s already in simplest form. √13 stays √13, because 13 has no factors other than 1 and itself, and 1 contributes nothing.

Does this work with variables under the radical?

The same pairing logic applies to exponents. √(x⁵) = √(x⁴ × x) = x²√x, since x⁴ is a perfect square. Odd exponents always leave one factor behind inside the radical.

Final thought

Simplifying radicals is a factoring skill wearing a square root costume. Get comfortable spotting perfect squares up to 100 and the rest of the method takes care of itself — use a checker to confirm, not to skip the practice.

Try the free Square Root Simplifier