Math calculator
Simplest radical form for any index and any coefficient, prime by prime.
Any index, any coefficient.
3√72
18√2
Approximately 8.485281374238.
Simplest form
18√2
index 2
Came out
6
× the coefficient 3 = 18
Stayed under
2
no complete group of the index
Largest power inside
36 = 6^2
the one-move route
Prime by prime
| Prime | Times inside | Groups of 2 | Comes out | Stays in |
|---|---|---|---|---|
| 2 | 3 | 1 | 2 | 2 |
| 3 | 2 | 1 | 3 | — |
A root of index 2 releases one copy of a prime for every 2 inside. Anything short of a complete group stays where it is.
If it ends up in a denominator
1 ÷ (18√2) = √2 / 36
A radical in a denominator is not wrong, but it is not simplest form either. Multiply top and bottom by √2: the bottom becomes 2 times what it was — 36 — and the root moves upstairs.
Every rationalised form on this page is checked numerically against the original in the test suite.
The coefficient multiplies whatever comes OUT from under the root and leaves what stays IN untouched. 3√72 is 18√2, not 18√216 — and that step is where the arithmetic usually goes wrong.
Exact; every rationalisation checked numerically.
Updated 7 September 2026 · Works in any browser, no installation
Pull out every complete group of the index, then multiply what came out by the coefficient. For 3√72: 72 is 2³ × 3², so a 2 and a 3 come out as 6 and a single 2 stays. The 6 multiplies the 3 in front to give 18, and the answer is 18√2. What stayed under the root is never touched by the coefficient — that is the step to check.
At a glance
This is the step that goes wrong. The coefficient multiplies what comes OUT of the root, not what stays in.
3√72 simplifies to 3 × 6√2 = 18√2. The 2 under the root is untouched. Multiplying it instead gives 18√216, which is a different and much larger number.
The reason is that the coefficient was never inside. 3√72 means 3 multiplied by √72; simplifying the root does not change what is multiplying it. The page shows the multiplication explicitly for that reason.
A root of index k releases one copy of a prime for every k of them under the sign. Square roots need pairs, cube roots need triples, fourth roots need groups of four.
Anything short of a complete group stays. 2³ under a cube root gives one 2 out and nothing left; 2³ under a square root gives one 2 out and one 2 still under.
The table on this page shows that division for each prime: how many times it appears, how many complete groups that makes, and what the remainder is. Once it is laid out that way there is nothing left to remember.
The prime route is the reliable one: factorise, divide each exponent by the index, done. It always works and it never needs a guess.
The other route is to spot the largest perfect power that divides the radicand and split it off in one move. For 72 that is 36, so √72 = √36 × √2 = 6√2. Faster when you see it, and easy to under-shoot — splitting off 4 instead of 36 gives 2√18, which is correct but not finished.
This page gives both, so the one-move split can be checked against the prime route rather than trusted.
1/√2 and √2/2 are the same number. The second is the conventional form, and getting there is called rationalising the denominator.
Multiply top and bottom by √2: the bottom becomes 2 and the root moves upstairs. Nothing is lost, because multiplying by √2/√2 is multiplying by 1.
The convention predates calculators — dividing by 1.414 by hand is much harder than dividing by 2 — but it survives because it gives a canonical form, so two people arrive at the same expression. For a cube root you need TWO more copies to complete the group, not one, which is the part people miss.
Simplest radical form means no factor under the root is a perfect power of the index. It is a canonical form: any correct route reaches the same expression.
That is what makes surds addable. √8 + √18 looks like nothing until both are simplified — 2√2 + 3√2 — and then it is obviously 5√2. Unsimplified, the two terms do not visibly match.
It is also what makes answers checkable. Two students who simplify correctly write the same thing, which is precisely why the form is required rather than merely suggested.
√15 is already in simplest form. 15 is 3 × 5, no prime appears twice, and nothing can be released.
That is a positive result and this page says so explicitly. A tool that echoes the input back without comment leaves it ambiguous whether it simplified and found nothing or failed to run.
Numbers with nothing to pull out are called squarefree, and most numbers are: about 61% of them. So “already simplest” is the common case rather than the exception.
Simplest radical form is a convention with a definition; these set it out.
Method. The radicand is factorised and each prime is divided by the index: the quotient is how many copies come out, the remainder is what stays. The coefficient is then multiplied into the outside part only. Rationalised forms are checked numerically in the suite against the original expression, so a wrong numerator would fail rather than merely look plausible. That engine is verified on every change against 68 hand-written assertions, including that the coefficient multiplies only the outside part across twenty-four hundred cases, and that every simplified radical reconstructs its radicand exactly. The count and the per-case breakdown are published on the formula verification page.
If you only want a square root with its decimal, the Square Root Calculator is the shorter page. Cube roots of negatives — which are real — are on the Cube Root Calculator.
Educational use disclaimer
This is an educational tool. Simplest radical form is a canonical form, so the answer here is the one any correct method reaches — but your course may present the working in a different order.
How we calculate · Found an error? email us
Published 7 September 2026