Square Root Calculator › Square root of 10
Square Root of 10
√10 ≈ 3.162278
10 is not a perfect square, so √10 is an irrational number that is approximately 3.162278. In simplified radical form, √10 = √10.
√10 lies between √9 = 3 and √16 = 4.
| Number | 10 |
| Square root (decimal) | 3.162278 |
| Simplified radical | √10 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 10
10 = 2 × 5. This is what determines the simplified radical: for √10 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √10 is already in simplest radical form as √10.
Estimating √10 by hand
√10 falls between √9 = 3 and √16 = 4, so it starts at 3.something. The Babylonian method refines that guess by averaging it with 10 ÷ the guess:
| Step | Estimate |
| Start | 3 |
| Step 1 | 3.166667 |
| Step 2 | 3.162281 |
| Step 3 | 3.162278 |
After 3 steps the estimate is 3.162278, matching √10 ≈ 3.162278.
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Frequently asked questions
Is 10 a perfect square?
No, 10 is not a perfect square, so its square root is irrational and approximately 3.162278.
What is the square root of 10 in simplest radical form?
√10 = √10 in simplest radical form.