Square root calculator that gives the exact answer, then the decimal
Enter a number and get its square root in simplest radical form, with the perfect-square factor named and pulled out in front of you. The decimal comes second, because a decimal is an approximation and the radical is the actual value.
Handles cube roots too, and says plainly when a number will not simplify. Part of SolveStep's algebra tools.
Why √180 is answered as 6√5 and not 13.4164
A decimal is an approximation. 13.4164 is not the square root of 180; it is a number very close to it. 6√5 is the square root of 180, exactly, with nothing rounded away.
That matters more than it first appears. Carry 6√5 through a longer calculation and terms cancel cleanly; carry 13.4164 and the rounding error compounds at every step. It is also what an algebra or geometry exam expects when the instruction says "leave your answer in simplest radical form", which is why the phrase appears on so many mark schemes.
Simplifying means pulling out the largest perfect square hiding inside the number. 180 = 36 × 5, and 36 is 6², so the 6 comes outside and the 5 stays under the radical. What is left inside is square-free: no perfect square divides it any more, which is the test for being fully simplified.
A perfect square
√144 = 12
Some numbers come out whole. 12² = 144, so no radical remains at all.
A square factor to pull out
√180 = 6√5
180 = 36 × 5. The 36 is 6², so the 6 steps outside and the 5 stays in.
Already simplest
√65 = √65
65 = 5 × 13, and neither is a square, so nothing can come out. The exact answer is the radical itself.
A cube root
³√54 = 3³√2
For cube roots you pull out perfect cubes instead. 54 = 27 × 2, and 27 is 3³.
How to simplify a square root by hand
If the number is 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and so on, the root is a whole number and you are finished. Knowing the squares up to 20² = 400 by heart saves a lot of time.
Work down from the largest square below your number. For 180: is it divisible by 169? No. 144? No. 121? No. 100? No. 81? No. 64? No. 49? No. 36? Yes, 180 = 36 × 5. Stop there.
√180 = √(36 × 5) = √36 × √5. This split is legal because the square root of a product equals the product of the square roots, for non-negative numbers.
√36 = 6, so the expression becomes 6√5. The 6 is outside the radical, the 5 stays inside.
5 is prime, so nothing more can come out. If you had stopped at 180 = 4 × 45 and written 2√45, the answer would be correct but not simplified, because 45 still contains 9.
The three cases you will meet
A number with a large square factor
√180- Not a perfect square: 13² = 169 and 14² = 196, so the root is between 13 and 14.
- Work down through the squares. 36 divides 180, and 180 ÷ 36 = 5.
- Split: √180 = √36 · √5 = 6√5.
- 5 is prime, so nothing further comes out. As a decimal, 6√5 ≈ 13.4164.
A number that will not simplify
√65- 65 = 5 × 13, and both factors are prime.
- No perfect square other than 1 divides 65, so there is nothing to pull out.
- The exact answer is √65 itself. That is a complete answer, not an unfinished one.
- As a decimal it is about 8.0623, which is close to 8 because 64 is the nearest perfect square.
Stopping too early
√72- A common first attempt: 72 = 4 × 18, giving 2√18.
- That is true but not simplified, because 18 still contains the square factor 9.
- Continue: 2√18 = 2 · 3√2 = 6√2.
- Going straight to the largest square factor avoids the second round: 72 = 36 × 2 gives 6√2 in one step.
Why negative numbers have no real square root
Squaring any real number gives a non-negative result: a positive times a positive is positive, and a negative times a negative is also positive. So no real number squares to give −9, and √−9 has no real value.
Mathematics answers this by inventing one. The imaginary unit i is defined so that i² = −1, which makes √−9 = 3i. This calculator says so rather than returning an error, because knowing the complex answer exists is usually what the question is really testing.
Where simplified radicals turn up
Constantly in geometry. The diagonal of a unit square is √2; the height of an equilateral triangle with side 2 is √3; the distance between two points comes out of the distance formula as a root that usually wants simplifying.
They also appear whenever a quadratic has irrational roots. x² − 5x + 5 = 0 has discriminant 5, so the roots are (5 ± √5)/2, and the quadratic equation solver leaves them in that exact form for the same reason this page does.
Square root calculator FAQ
What is the square root of 180 in simplest radical form?
6√5. Since 180 = 36 × 5 and 36 is a perfect square, the 6 comes outside the radical and the 5 stays inside. As a decimal it is approximately 13.4164, but 6√5 is the exact value.
What is the square root of 65?
√65, which is already in simplest form. 65 factors as 5 × 13 and neither is a perfect square, so nothing can be taken outside the radical. Its decimal value is approximately 8.0623.
How do you simplify a square root?
Find the largest perfect square that divides the number, split the radical into that square times the rest, and take the root of the square part outside. For √72: 72 = 36 × 2, so √72 = √36 · √2 = 6√2. What remains inside must have no perfect square factors left.
Why leave an answer as a radical instead of a decimal?
Because the radical is exact and the decimal is not. 6√5 is the square root of 180; 13.4164 is only close to it. In a longer calculation, exact radicals cancel cleanly while rounded decimals accumulate error, which is why exams ask for simplest radical form.
Can you take the square root of a negative number?
Not within the real numbers, because squaring any real number gives a non-negative result. In the complex numbers, √−9 = 3i, where i is defined by i² = −1. This calculator says so rather than simply refusing.
How do you simplify a cube root?
The same way, but pulling out perfect cubes rather than perfect squares. ∛54 = ∛(27 × 2) = 3∛2, because 27 = 3³. Unlike square roots, cube roots of negative numbers do exist in the reals: ∛−8 = −2.
Is this square root calculator free?
Yes, free with no account and no limit. It gives the exact simplified radical first and the decimal second, and names the square factor it pulled out so you can follow the method rather than only read the result.
Tools that go with radicals
Quadratic Equation Solver
Irrational roots left in exact radical form, not rounded.
Factoring Calculator
Factor a quadratic, with the pattern that was used named.
Slope Calculator
Distance between two points, which usually lands on a radical.
Fraction Calculator
Exact fraction arithmetic, reduced rather than rounded.
Scientific Notation
Very large and very small numbers written the compact way.