Square Root Calculator › Square root of 31
Square Root of 31
√31 ≈ 5.567764
31 is not a perfect square, so √31 is an irrational number that is approximately 5.567764. In simplified radical form, √31 = √31.
√31 lies between √25 = 5 and √36 = 6.
| Number | 31 |
| Square root (decimal) | 5.567764 |
| Simplified radical | √31 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 31
31 = 31. This is what determines the simplified radical: for √31 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √31 is already in simplest radical form as √31.
Estimating √31 by hand
√31 falls between √25 = 5 and √36 = 6, so it starts at 5.something. The Babylonian method refines that guess by averaging it with 31 ÷ the guess:
| Step | Estimate |
| Start | 5 |
| Step 1 | 5.6 |
| Step 2 | 5.567857 |
| Step 3 | 5.567764 |
After 3 steps the estimate is 5.567764, matching √31 ≈ 5.567764.
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Frequently asked questions
Is 31 a perfect square?
No, 31 is not a perfect square, so its square root is irrational and approximately 5.567764.
What is the square root of 31 in simplest radical form?
√31 = √31 in simplest radical form.