Square Root Calculator › Square root of 174
Square Root of 174
√174 ≈ 13.190906
174 is not a perfect square, so √174 is an irrational number that is approximately 13.190906. In simplified radical form, √174 = √174.
√174 lies between √169 = 13 and √196 = 14.
| Number | 174 |
| Square root (decimal) | 13.190906 |
| Simplified radical | √174 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 174
174 = 2 × 3 × 29. This is what determines the simplified radical: for √174 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √174 is already in simplest radical form as √174.
Estimating √174 by hand
√174 falls between √169 = 13 and √196 = 14, so it starts at 13.something. The Babylonian method refines that guess by averaging it with 174 ÷ the guess:
| Step | Estimate |
| Start | 13 |
| Step 1 | 13.192308 |
| Step 2 | 13.190906 |
| Step 3 | 13.190906 |
After 3 steps the estimate is 13.190906, matching √174 ≈ 13.190906.
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Frequently asked questions
Is 174 a perfect square?
No, 174 is not a perfect square, so its square root is irrational and approximately 13.190906.
What is the square root of 174 in simplest radical form?
√174 = √174 in simplest radical form.