Square Root Calculator › Square root of 178
Square Root of 178
√178 ≈ 13.341664
178 is not a perfect square, so √178 is an irrational number that is approximately 13.341664. In simplified radical form, √178 = √178.
√178 lies between √169 = 13 and √196 = 14.
| Number | 178 |
| Square root (decimal) | 13.341664 |
| Simplified radical | √178 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 178
178 = 2 × 89. This is what determines the simplified radical: for √178 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √178 is already in simplest radical form as √178.
Estimating √178 by hand
√178 falls between √169 = 13 and √196 = 14, so it starts at 13.something. The Babylonian method refines that guess by averaging it with 178 ÷ the guess:
| Step | Estimate |
| Start | 13 |
| Step 1 | 13.346154 |
| Step 2 | 13.341665 |
| Step 3 | 13.341664 |
After 3 steps the estimate is 13.341664, matching √178 ≈ 13.341664.
Nearby square roots
Frequently asked questions
Is 178 a perfect square?
No, 178 is not a perfect square, so its square root is irrational and approximately 13.341664.
What is the square root of 178 in simplest radical form?
√178 = √178 in simplest radical form.