Square Root Calculator › Square root of 177
Square Root of 177
√177 ≈ 13.304135
177 is not a perfect square, so √177 is an irrational number that is approximately 13.304135. In simplified radical form, √177 = √177.
√177 lies between √169 = 13 and √196 = 14.
| Number | 177 |
| Square root (decimal) | 13.304135 |
| Simplified radical | √177 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 177
177 = 3 × 59. This is what determines the simplified radical: for √177 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √177 is already in simplest radical form as √177.
Estimating √177 by hand
√177 falls between √169 = 13 and √196 = 14, so it starts at 13.something. The Babylonian method refines that guess by averaging it with 177 ÷ the guess:
| Step | Estimate |
| Start | 13 |
| Step 1 | 13.307692 |
| Step 2 | 13.304135 |
| Step 3 | 13.304135 |
After 3 steps the estimate is 13.304135, matching √177 ≈ 13.304135.
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Frequently asked questions
Is 177 a perfect square?
No, 177 is not a perfect square, so its square root is irrational and approximately 13.304135.
What is the square root of 177 in simplest radical form?
√177 = √177 in simplest radical form.