The exact square root first, then the decimal to as many places as you want.
Square root, exactly
Surd first, decimal second.
√72
6√2
8.485281374238 to 12 places, and the decimal never ends.
Exact form
6√2
simplified
Decimal
8.485281374238
irrational — this is a truncation
Both roots
±6√2
√ means the positive one; x² = n gives both
Prime factors
2³ × 3²
the pairs are what come out
The estimate to make first
72 sits between 64 and 81, so its square root is between 8 and 9 — which is the estimate to make before reaching for a calculator, and the check to apply afterwards.
How the decimal was found
Newton’s method converging on the square root, guess by guess
Step
Guess
Next guess
1
100.00000000
50.36000000
2
50.36000000
25.89485305
3
25.89485305
14.33766418
4
14.33766418
9.67970133
5
9.67970133
8.55897368
6
8.55897368
8.48559861
7
8.48559861
8.48528138
Newton’s method: average the guess with the number divided by the guess. It roughly doubles the correct digits each step, and it runs on whole numbers here so the digits are real rather than a double’s.
2³ × 3² contains a pair of 2s and a pair of 3s. Each pair comes out as a single copy, which is why the 6 is outside and the 2 stays in.
A square root that is not a whole number is irrational — it cannot be written as any fraction at all. The decimal below never terminates and never repeats, so every printed version of it is a truncation.
Every positive number has TWO square roots, 6√2 and −6√2, because both square to 72. The √ symbol means the positive one specifically, which is why solving x² = 72 gives ±6√2 while √72 gives just one.
The exact form comes from the prime factorisation; the decimal is computed separately on integers.
What this tool shows
√72 is 6√2, and that is the answer a marker wants — exact, and the form that goes on to combine with other surds. The decimal comes second, computed on whole numbers so its digits are real past where a double stops.
The exact square root in simplest surd form
The decimal to up to sixty places
Whether the number is a perfect square
Negative radicands, answered in terms of i
Both roots, positive and negative
The two perfect squares it sits between
Exact surd, not just a decimal Integer Newton, sixty places Negatives answered in i The estimate to make first
Updated 7 September 2026 · Works in any browser, no installation
To simplify a square root, factorise what is under it and pull out every PAIR of equal primes. 72 is 2³ × 3², which holds a pair of 2s and a pair of 3s, so a 2 and a 3 come out as 6 and a single 2 stays behind: √72 = 6√2. If nothing pairs up, nothing comes out and the root is already in simplest form — which is an answer, not a failure.
At a glance
Formula shown
\u221an simplifies by pulling out every pair of equal primes: \u221a(2\u00b3 \u00d7 3\u00b2) = 2 \u00d7 3 \u00d7 \u221a2 = 6\u221a2. The decimal comes from Newton iteration on integers: x \u2192 (x + n/x) / 2, which roughly doubles the correct digits each step.
Scenario support
Giving an exact answer where a decimal was only an approximation; simplifying a surd before combining it with another; finding a side length from an area.
Educational estimate
Planning support from the values you enter — not professional advice.
What comes out from under the sign
A square root releases one copy of a prime for every TWO under the sign. That is the whole rule, and it is why factorising first makes the answer obvious.
72 = 2 × 2 × 2 × 3 × 3. Two of the 2s pair up and come out as a single 2; the two 3s come out as a 3. The leftover 2 has no partner and stays. So 6√2.
√15 is 3 × 5 — no prime appears twice, so nothing comes out. The tool says so explicitly rather than echoing the input back, because “already simplest” is the answer to that question.
Why there are always two
Every positive number has two square roots. 6√2 squares to 72, and so does −6√2, because a negative times a negative is positive.
The √ symbol means the POSITIVE one specifically. That is a convention, and it exists so that √ is a function — one input, one output. Without it, every expression containing a root would be ambiguous.
Which is why solving x² = 72 gives ±6√2 while √72 gives 6√2 alone. Losing the ± when solving an equation is the single commonest mistake with roots, and it loses exactly half the answers.
Estimating without a calculator
Before reaching for anything, find the two perfect squares your number sits between. 72 is between 64 and 81, so its root is between 8 and 9 — and closer to 8.5, since 72 is nearer the middle.
That takes a few seconds and catches every order-of-magnitude error. A tool that returns 84.85 instead of 8.485 is obviously wrong once you have the bracket, and obviously plausible without it.
The page states the bracket every time for exactly that reason. It is the check to apply to the answer, not a substitute for it.
Irrational, and what that means
If a whole number is not a perfect square, its root is irrational: no fraction equals it, and the decimal never terminates and never repeats.
That is a stronger statement than “the decimal is long”. √2 cannot be written as a/b for any whole numbers at all, and the proof is short: if it could, and the fraction were in lowest terms, both numerator and denominator would have to be even — contradicting “lowest terms”.
So every printed decimal for √2 is a truncation, including the sixty places this page will give you. The surd form is the only exact way to write it, which is the practical argument for keeping answers in surds until the last step.
The square root of a negative
No real number squares to a negative, so √−9 has no real value. Some tools return an error, which is true but unhelpful.
In the complex numbers it has a value: i is defined as the square root of −1, so √−9 is 3i, and √−72 is 6i√2. The simplification works exactly as before — the i comes along for the ride.
The check is the same as always: (3i)² = 9i² = 9 × (−1) = −9. This page gives that answer rather than refusing, because “no real value” and “no value” are different statements.
How the decimal is actually found
Newton’s method: guess, then replace the guess with the average of it and the number divided by it. Repeat. If the guess is too high, n/x is too low, and the average lands between them.
It converges quadratically, which means the number of correct digits roughly DOUBLES each step. Six or seven steps is enough for the sixty places this page will print, from almost any starting guess.
Two details make the digits here trustworthy. The iteration runs on whole numbers scaled by a power of ten rather than on floating point, so nothing is lost past the seventeenth digit; and the result is checked by raising it back — the truncated root must be below the radicand and one unit more must be above it.
Sources and methodology
Surd simplification and Newton iteration are both classical. These are the references.
Method. The radicand is factorised into primes and every complete pair is pulled out, which is what produces the exact surd. The decimal is a separate computation: Newton iteration carried out on integers scaled by a power of ten, so the digits reported are the true ones rather than a double re-formatted. Both are checked in the suite by squaring the answer back — the surd exactly, the decimal to within one unit in the last place. That engine is verified on every change against 68 hand-written assertions, including that every simplified radical to five thousand multiplies back to its radicand exactly, and that each truncated decimal brackets its radicand when raised back. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Cube RootCube roots exactly, negatives included — the cube root of minus 27 is minus 3, a real answer — with the two complex roots named alongside.
Simplify RadicalsSimplest radical form at any index from 2 to 12 with a coefficient in front, shown prime by prime, plus the rationalised denominator.
ExponentPowers with the awkward cases right — a negative exponent is a reciprocal not a sign, a fractional one is a root, and zero to the zero is reported as contested.
Prime FactorizationBreak any number into primes with the division ladder shown, the number of trial divisions reported, and the argument for why the search can stop at the square root.
LogLogarithms in any base with the exponential form beside them, whole answers confirmed by raising the base back, and change of base worked through.
Quadratic FormulaSolve any quadratic with exact roots — surds stay surds and a negative discriminant gives the complex pair — plus the vertex, the factored form and every step of the working.
For a coefficient in front, or any index other than two, the Simplify Radicals Calculator handles the general case. For cube roots specifically — where a negative has a real answer — the Cube Root Calculator is the page.
Educational use disclaimer
This is an educational tool. The surd form is exact; the decimal is a truncation of an irrational number wherever the root is not whole, and the page says which case you are in.
Published the square root page with the exact surd first and the decimal second, because 6 root 2 is the answer a marker wants and 8.485281 is not.
The decimal comes from Newton iteration carried out on integers scaled by a power of ten, so digits past the seventeenth are real rather than a double reformatted.
A negative radicand is answered in terms of i rather than refused, since 'no real value' and 'no value' are different statements.
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