Simplest Radical Form: How to Simplify a Square Root

Most calculators answer the square root of 72 with 8.485281374 and stop there. That is fine if you are cutting timber. It is no use at all when the question says give your answer in simplest radical form, which is a polite way of saying do not round anything.

What simplest radical form is asking for

Three conditions have to hold at once. No perfect square factor may remain under the radical sign. No radical may sit in a denominator. No fraction may sit under a radical. Satisfy all three and the expression is finished.

The square root of 72 fails the first test, because 36 divides into 72 and 36 is a perfect square. Split it: root 72 equals root 36 times root 2, which is 6 root 2. Nothing square is left inside, since 2 has no factors to speak of. Check the decimals if you want reassurance: 6 times 1.41421 is 8.4853, which matches the plain square root.

When the perfect square is not obvious

Root 48 looks awkward until you spot that 16 goes into it: root 48 is root 16 times root 3, so 4 root 3, about 6.928. Root 200 splits as root 100 times root 2, giving 10 root 2, about 14.142. When nothing jumps out, break the number into prime factors and pair them up. 180 factors to 2 times 2 times 3 times 3 times 5. Each pair escapes as a single number, so the twos give a 2, the threes give a 3, and 2 times 3 is 6 out front with the lone 5 left inside: 6 root 5.

Pull out the largest square, not the first one you notice. Treating root 72 as root 4 times root 18 gives 2 root 18, which is correct but unfinished, because 18 still contains 9. Simplify again and you land back on 6 root 2.

Why the exact form earns its extra step

Radicals only combine when the part under the sign matches. Root 8 plus root 18 looks untouchable until you simplify both: 2 root 2 plus 3 root 2, which adds to 5 root 2, about 7.071. In decimals that is 2.828 plus 4.243, the same total with no way to see the answer is exactly 5 root 2.

Rounding also drifts. Multiply 8.485 by itself and you get 71.995, not 72.

Using the square root simplifier

Type the number under the radical into the input box and press calculate. The tool returns the simplified radical, the decimal value and the factor pair it used, so you can see why the answer looks the way it does.

It handles a coefficient in front, so 3 root 50 comes back as 15 root 2. It handles a fraction under the radical, and it will rationalise a denominator for you.

Getting a radical out of a denominator

3 over root 5 is not finished, because the radical is downstairs. Multiply top and bottom by root 5, which changes nothing at all because you are multiplying by 1: you get 3 root 5 over 5, roughly 1.342. Teachers still ask for this even though a calculator does not care, and it is worth doing, because the tidied form is far easier to combine with other terms later. The same reduce-to-lowest-terms instinct shows up when you simplify fractions.

Common questions

Is 6 root 2 really the same as root 72? Yes. They are two spellings of one number. 6 root 2 is exact and 8.485 is that number cut off after three decimals. If the answer feeds into more arithmetic, keep the radical, because rounding early compounds the error.

Which numbers cannot be simplified? Any whose prime factors all appear once. Root 2, root 3, root 5, root 6, root 7, root 10 and root 11 are already in simplest form. Root 30 stays as it is too, because 30 is 2 times 3 times 5 with no repeats.

What if I only want the decimal? Then use the plain square root calculator instead. The simplifier exists for exact answers; the standard tool is quicker when you want a number to type into a spreadsheet.

Does the same method work for cube roots? The idea does, but the grouping changes. For cube roots you pull out factors that appear three times rather than twice, so the cube root of 54 is 3 times the cube root of 2.