Square Root Calculator › Square root of 51
Square Root of 51
√51 ≈ 7.141428
51 is not a perfect square, so √51 is an irrational number that is approximately 7.141428. In simplified radical form, √51 = √51.
√51 lies between √49 = 7 and √64 = 8.
| Number | 51 |
| Square root (decimal) | 7.141428 |
| Simplified radical | √51 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 51
51 = 3 × 17. This is what determines the simplified radical: for √51 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √51 is already in simplest radical form as √51.
Estimating √51 by hand
√51 falls between √49 = 7 and √64 = 8, so it starts at 7.something. The Babylonian method refines that guess by averaging it with 51 ÷ the guess:
| Step | Estimate |
| Start | 7 |
| Step 1 | 7.142857 |
| Step 2 | 7.141429 |
| Step 3 | 7.141428 |
After 3 steps the estimate is 7.141428, matching √51 ≈ 7.141428.
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Frequently asked questions
Is 51 a perfect square?
No, 51 is not a perfect square, so its square root is irrational and approximately 7.141428.
What is the square root of 51 in simplest radical form?
√51 = √51 in simplest radical form.