Square Root Calculator › Square root of 113
Square Root of 113
√113 ≈ 10.630146
113 is not a perfect square, so √113 is an irrational number that is approximately 10.630146. In simplified radical form, √113 = √113.
√113 lies between √100 = 10 and √121 = 11.
| Number | 113 |
| Square root (decimal) | 10.630146 |
| Simplified radical | √113 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 113
113 = 113. This is what determines the simplified radical: for √113 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √113 is already in simplest radical form as √113.
Estimating √113 by hand
√113 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 113 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.65 |
| Step 2 | 10.630164 |
| Step 3 | 10.630146 |
After 3 steps the estimate is 10.630146, matching √113 ≈ 10.630146.
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Frequently asked questions
Is 113 a perfect square?
No, 113 is not a perfect square, so its square root is irrational and approximately 10.630146.
What is the square root of 113 in simplest radical form?
√113 = √113 in simplest radical form.