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?

This is a real square-root simplifier. A negative radicand is refused; complex roots, variables, cube roots, sums, products of several radicals and radical equations are outside this workflow. Choose Fraction for a nonnegative integer fraction under the root; a decimal radicand needs an exact integer-fraction rewrite.

How large a radicand can it simplify?

Whole-number radicands can be at most 1,000,000,000,000. In Fraction mode the reduced numerator times denominator must fit that same limit, and the coefficient after rewriting must have a reduced top and bottom at most 1,000,000,000. Values beyond the limits get an explicit message. Factoring time depends on the input and your device.

Is the decimal exact?

A terminating rational result has an exact decimal. A remaining radical or nonterminating rational has a rounded preview marked ≈, with ten significant digits: √72 shows ≈ 8.485281374, and 1/3 shows ≈ 0.3333333333. The radical or rational fraction is the exact answer. Copy includes the same status.

Is anything uploaded, and is my work kept?

The calculation runs in this browser without uploading your inputs. The whole-number radicand and coefficient fields are remembered locally when you change them; the Fraction draft and mode stay in memory for this visit. The page opens in Whole number mode. Copy exports the original expression, simplified answer, decimal status and numbered steps.

How do I simplify a fraction under a square root?

Choose Fraction and enter integer numerator/denominator, such as 8/9. After reducing a/b, the calculator uses √(a/b) = √(a × b)/b and simplifies the resulting radical, giving a rational denominator. For √(8/9), √72/9 becomes 2√2/3. √(50/8) reduces to √(25/4) and gives 5/2 exactly. A coefficient such as 2/3 is applied to the result.

Does this simplify variable expressions or solve radical equations?

It simplifies one supplied real numeric square root and coefficient, including an integer fraction under the root. It does not infer variable signs or domains, combine sums of radicals, simplify cube or higher roots, or solve an equation for an unknown.

Radical Simplifier

Enter a whole number or choose Fraction for a root such as √(8/9). The calculator extracts perfect square factors, applies an optional whole, fraction or decimal coefficient, and shows the exact answer with numbered steps. The decimal preview is marked when it is approximate.

Simplified6√2√72 · 72 = 36 × 2 · ≈ 8.485281374

Radicand

Coefficient

Simplified

6√2

Outside

Inside

Decimal

≈ 8.485281374

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

72 = 2^3 × 3^2.
The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2.
√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?: This is a real square-root simplifier. A negative radicand is refused; complex roots, variables, cube roots, sums, products of several radicals and radical equations are outside this workflow. Choose Fraction for a nonnegative integer fraction under the root; a decimal radicand needs an exact integer-fraction rewrite.
How large a radicand can it simplify?: Whole-number radicands can be at most 1,000,000,000,000. In Fraction mode the reduced numerator times denominator must fit that same limit, and the coefficient after rewriting must have a reduced top and bottom at most 1,000,000,000. Values beyond the limits get an explicit message. Factoring time depends on the input and your device.
Is the decimal exact?: A terminating rational result has an exact decimal. A remaining radical or nonterminating rational has a rounded preview marked ≈, with ten significant digits: √72 shows ≈ 8.485281374, and 1/3 shows ≈ 0.3333333333. The radical or rational fraction is the exact answer. Copy includes the same status.
Is anything uploaded, and is my work kept?: The calculation runs in this browser without uploading your inputs. The whole-number radicand and coefficient fields are remembered locally when you change them; the Fraction draft and mode stay in memory for this visit. The page opens in Whole number mode. Copy exports the original expression, simplified answer, decimal status and numbered steps.
How do I simplify a fraction under a square root?: Choose Fraction and enter integer numerator/denominator, such as 8/9. After reducing a/b, the calculator uses √(a/b) = √(a × b)/b and simplifies the resulting radical, giving a rational denominator. For √(8/9), √72/9 becomes 2√2/3. √(50/8) reduces to √(25/4) and gives 5/2 exactly. A coefficient such as 2/3 is applied to the result.
Does this simplify variable expressions or solve radical equations?: It simplifies one supplied real numeric square root and coefficient, including an integer fraction under the root. It does not infer variable signs or domains, combine sums of radicals, simplify cube or higher roots, or solve an equation for an unknown.

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.