Square Root Calculator › Square root of 157
Square Root of 157
√157 ≈ 12.529964
157 is not a perfect square, so √157 is an irrational number that is approximately 12.529964. In simplified radical form, √157 = √157.
√157 lies between √144 = 12 and √169 = 13.
| Number | 157 |
| Square root (decimal) | 12.529964 |
| Simplified radical | √157 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 157
157 = 157. This is what determines the simplified radical: for √157 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √157 is already in simplest radical form as √157.
Estimating √157 by hand
√157 falls between √144 = 12 and √169 = 13, so it starts at 12.something. The Babylonian method refines that guess by averaging it with 157 ÷ the guess:
| Step | Estimate |
| Start | 12 |
| Step 1 | 12.541667 |
| Step 2 | 12.52997 |
| Step 3 | 12.529964 |
After 3 steps the estimate is 12.529964, matching √157 ≈ 12.529964.
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Frequently asked questions
Is 157 a perfect square?
No, 157 is not a perfect square, so its square root is irrational and approximately 12.529964.
What is the square root of 157 in simplest radical form?
√157 = √157 in simplest radical form.