About the Square Root calculator
This square root calculator finds square roots, cube roots and 4th, 5th and 6th roots of any number. For whole numbers it also gives the simplified radical form, so instead of just 8.485 you see that √72 = 6√2. It tells you whether the number is a perfect square or cube and checks the answer by raising it back to the power.
The working uses prime factorization with your own number, which is exactly the method taught in school for simplifying radicals. Negative numbers are handled too: odd roots give a negative answer and even roots give an imaginary one.
How to use it
- Type your number into Number.
- Choose the Root: square root, cube root, or a 4th, 5th or 6th root.
- Read the decimal answer and, for whole numbers, the simplified radical form.
- Look at the steps to see the prime factors grouped and pulled out of the root.
The formula
The nth root of x is the number r with rn = x. So √49 = 7 because 7² = 49, and ∛125 = 5 because 5³ = 125. It can also be written as a power: n√x = x1/n.
To simplify a radical, write the number as a product of primes and take out every complete group of n equal factors:
√72 = √(2³ × 3²) = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2
- x — the number under the root (the radicand)
- n — the index of the root: 2 for square, 3 for cube
- √(−x) — for even n and negative x, the answer is imaginary:
√(−16) = 4i
Simplifying radicals step by step
Take √180. Break it into primes: 180 = 2² × 3² × 5. For a square root you look for pairs. There is a pair of 2s and a pair of 3s, so a 2 and a 3 come out, while the single 5 stays inside: √180 = 2 × 3 × √5 = 6√5.
Cube roots work the same way with groups of three. 54 = 2 × 3³, so ∛54 = 3∛2.
A shortcut is to spot the largest perfect square factor. 180 = 36 × 5, and √36 = 6, giving 6√5 straight away. Knowing the first few perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) makes this fast. For the prime breakdown of a larger number, use the prime factorization calculator.
Estimating a square root without a calculator
Find the two perfect squares either side of your number. For √50, 49 < 50 < 64, so the answer is between 7 and 8, and very close to 7 because 50 is just above 49.
To get closer, use √x ≈ a + (x − a²) ÷ 2a, where a² is the nearest perfect square. For √50, a = 7, so √50 ≈ 7 + 1/14 ≈ 7.071. The true value is 7.0711, so the estimate is excellent.
Remember that √(a + b) is not √a + √b. For example, √(9 + 16) = √25 = 5, not 3 + 4 = 7. That is the idea behind the Pythagorean theorem.
Frequently asked questions
What is the square root of a negative number?
There is no real square root of a negative number. The answer is imaginary, using i = √(−1). For example, √(−25) = 5i.
Can you take the cube root of a negative number?
Yes. Odd roots of negative numbers are real and negative. ∛(−27) = −3 because (−3)³ = −27.
How do I know if a number is a perfect square?
Its square root is a whole number. The calculator tells you, and in prime-factor form every exponent of a perfect square is even.
Why are there two square roots of a number?
Both 6 and −6 square to 36. The √ symbol means the principal, non-negative root, so √36 = 6. When solving x² = 36, you need both: x = ±6.
What is the square root of 2?
√2 ≈ 1.41421356237. It is irrational, so its decimal never ends or repeats, and √2 is already in simplest radical form.
Last updated October 24, 2026. Results are estimates for education and planning — see our disclaimer.