Square Root Calculator › Square root of 525
Square Root of 525
√525 ≈ 22.912878
525 is not a perfect square, so √525 is an irrational number that is approximately 22.912878. In simplified radical form, √525 = 5√21.
√525 lies between √484 = 22 and √529 = 23.
| Number | 525 |
| Square root (decimal) | 22.912878 |
| Simplified radical | 5√21 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 525
525 = 3 × 5^2 × 7. This is what determines the simplified radical: for √525 you pull out one copy of any prime that appears twice.
Here the paired factor 5 comes out from under the root, leaving √525 = 5√21.
Estimating √525 by hand
√525 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 525 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.931818 |
| Step 2 | 22.912886 |
| Step 3 | 22.912878 |
After 3 steps the estimate is 22.912878, matching √525 ≈ 22.912878.
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Frequently asked questions
Is 525 a perfect square?
No, 525 is not a perfect square, so its square root is irrational and approximately 22.912878.
What is the square root of 525 in simplest radical form?
√525 = 5√21 in simplest radical form.