Square Root Calculator › Square root of 57
Square Root of 57
√57 ≈ 7.549834
57 is not a perfect square, so √57 is an irrational number that is approximately 7.549834. In simplified radical form, √57 = √57.
√57 lies between √49 = 7 and √64 = 8.
| Number | 57 |
| Square root (decimal) | 7.549834 |
| Simplified radical | √57 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 57
57 = 3 × 19. This is what determines the simplified radical: for √57 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √57 is already in simplest radical form as √57.
Estimating √57 by hand
√57 falls between √49 = 7 and √64 = 8, so it starts at 7.something. The Babylonian method refines that guess by averaging it with 57 ÷ the guess:
| Step | Estimate |
| Start | 7 |
| Step 1 | 7.571429 |
| Step 2 | 7.549865 |
| Step 3 | 7.549834 |
After 3 steps the estimate is 7.549834, matching √57 ≈ 7.549834.
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Frequently asked questions
Is 57 a perfect square?
No, 57 is not a perfect square, so its square root is irrational and approximately 7.549834.
What is the square root of 57 in simplest radical form?
√57 = √57 in simplest radical form.