Square Root Calculator › Square root of 524
Square Root of 524
√524 ≈ 22.891046
524 is not a perfect square, so √524 is an irrational number that is approximately 22.891046. In simplified radical form, √524 = 2√131.
√524 lies between √484 = 22 and √529 = 23.
| Number | 524 |
| Square root (decimal) | 22.891046 |
| Simplified radical | 2√131 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 524
524 = 2^2 × 131. This is what determines the simplified radical: for √524 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √524 = 2√131.
Estimating √524 by hand
√524 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 524 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.909091 |
| Step 2 | 22.891053 |
| Step 3 | 22.891046 |
After 3 steps the estimate is 22.891046, matching √524 ≈ 22.891046.
Nearby square roots
Frequently asked questions
Is 524 a perfect square?
No, 524 is not a perfect square, so its square root is irrational and approximately 22.891046.
What is the square root of 524 in simplest radical form?
√524 = 2√131 in simplest radical form.