gizmobench

Radical Simplifier

Type a whole number under the root and read it back in simplest form, with the working that got there. The tool finds the largest perfect square that divides the radicand, takes its root outside and leaves a radicand with no repeated prime factor left in it, which is what “simplest form” means: √72 becomes 6√2 because 72 is 36 × 2. The prime factorization, the square that came out and the split are listed as numbered steps you can copy into an exercise book, and the number outside and the number inside are given as separate values as well as in the finished radical. A coefficient in front of the root is part of the job here rather than an afterthought, so a whole number, a fraction like 2/3 or a decimal like 1.5 is read exactly and reduced against what came out of the root. All of it is done in exact integer arithmetic on radicands up to 1,000,000,000,000, so the answer squares back to the number you typed rather than to a rounded version of it, and the decimal beside the radical is clearly marked as the approximation it is.

Simplified6√2√72  ·  72 = 36 × 2  ·  ≈ 8.485281374
Radicand
Coefficient
Simplified
6√2
Outside
6
Inside
2
Decimal
≈ 8.485281374

√72 = 6√2, because 72 = 36 × 2. That is about 8.485281374.

  1. 72 = 2^3 × 3^2.
  2. The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2.
  3. √72 = √36 × √2 = 6√2.

Whole numbers up to 1,000,000,000,000 under the root. The coefficient takes a whole number, a fraction like 2/3 or a decimal like 1.5, with a top and a bottom up to 1,000,000,000, and a blank field means 1. The radical is exact; the decimal beside it is rounded to ten significant digits and carries a ≈ when it is an approximation.

  • √7272 = 36 × 2
    6√2
  • √5050 = 25 × 2
    5√2
  • √13prime
    √13, already simplest
How a radical is simplified. Split the radicand into the largest perfect square that divides it and whatever is left, then take the root of that square outside: √72 is √36 × √2, and √36 is 6, so the answer is 6√2. Finding that square is why the prime factorization is the first step: pair the primes up, one of each pair comes out, and what is left underneath has no repeated prime at all, which is exactly what makes a radical simplest. Exact integer arithmetic: the largest perfect square is factored out and the steps are shown, so the simplified form is exactly equal to the original rather than a rounded stand-in. The decimal beside it is an approximation and says so.

Common questions

How do you simplify a square root?
Find the largest perfect square that divides the number under the root, split the number into that square times what is left, and take the square's root outside. For 72 the prime factorization is 2^3 × 3^2, the largest square factor is 36, and 72 = 36 × 2, so √72 = √36 × √2 = 6√2. The pairing is the quick way to see it: write the primes out, take one from each pair outside the root, and leave the unpaired ones underneath. This page shows those three lines as numbered steps for whatever you type, so the working is there and not just the answer.
When is a radical already in simplest form?
When no perfect square above 1 divides the number under the root, which is the same as saying no prime appears twice in its factorization. √13 is already simplest because 13 is prime, and the tool says so instead of pretending something came out. √30 is simplest too, even though 30 is not prime, because 30 is 2 × 3 × 5 with no repeats. Whatever you type, the number left under the root here is always square-free, so the answer cannot be simplified a second time.
Can I put a number or a fraction in front of the root?
Yes. The coefficient field takes a whole number, a fraction like 2/3, or a decimal like 1.5, and it is read as an exact fraction rather than as a floating point value, so 1.5 becomes 3/2 to the unit. It is then reduced against whatever came out of the root: 3√72 is 18√2, (2/3)√72 is 4√2 because the 3 cancels the 6 that came out, and (1/4)√72 is 3√2/2. A negative coefficient stays in front of the answer and in the decimal. Leave the field blank and it is read as 1, which is what the placeholder in the field shows.
Why will it not take a negative number under the root?
Because the square root of a negative number is not a real number, and this tool is declared as a real-number simplifier. Rather than invent a notation for you, it names the problem and points at the way through: type 72 instead of -72 to get 6√2, then write i beside that answer if you are working in complex numbers, since √-72 = 6i√2. Fractions and decimals under the root are refused in the same explicit way, with the rewrite that gets you back to a whole number.
How large a radicand can it simplify?
Up to 1,000,000,000,000, a million million, and above that it says so rather than hanging. The method is trial division to the square root of the radicand, which at the ceiling is about 266,000 divisions after the 2, 3, 5 wheel, measured at a few milliseconds; two digits higher would be ten times that on every keystroke. Everything under the ceiling is worked in exact integer arithmetic, so a twelve-digit radicand is factored exactly rather than through floating point, where a calculator starts rounding past 9,007,199,254,740,992.
Is the decimal exact?
Only when there is no radical left. If the root comes out whole, as with √144 = 12, or the coefficient leaves a fraction that terminates, the decimal is computed in exact integer arithmetic and shown as an equality. Whenever something is still under the root the value is irrational, so the decimal is rounded to ten significant digits and carries a ≈ in front of it: √72 shows ≈ 8.485281374. Copy the answer and the exported text says the same thing, marking the decimal as approximate. The simplified radical is the exact answer; the decimal is there to check the size.
Is anything uploaded, and is my work kept?
Nothing is uploaded. The factoring runs in your browser and the page makes no request while you type. The two fields are remembered in this browser alone so the page opens where you left it, including a draft that does not simplify yet, and the Start over button at the top of the page forgets them. The Copy button puts the whole answer on your clipboard as plain text: the original expression, the simplified radical, the decimal with its status, and the numbered steps.

Exact integer arithmetic: the largest perfect square is factored out and the steps are shown, so the simplified form is exactly equal to the original rather than a rounded stand-in. The decimal beside it is an approximation and says so.