Square Root Calculator › Square root of 154
Square Root of 154
√154 ≈ 12.409674
154 is not a perfect square, so √154 is an irrational number that is approximately 12.409674. In simplified radical form, √154 = √154.
√154 lies between √144 = 12 and √169 = 13.
| Number | 154 |
| Square root (decimal) | 12.409674 |
| Simplified radical | √154 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 154
154 = 2 × 7 × 11. This is what determines the simplified radical: for √154 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √154 is already in simplest radical form as √154.
Estimating √154 by hand
√154 falls between √144 = 12 and √169 = 13, so it starts at 12.something. The Babylonian method refines that guess by averaging it with 154 ÷ the guess:
| Step | Estimate |
| Start | 12 |
| Step 1 | 12.416667 |
| Step 2 | 12.409676 |
| Step 3 | 12.409674 |
After 3 steps the estimate is 12.409674, matching √154 ≈ 12.409674.
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Frequently asked questions
Is 154 a perfect square?
No, 154 is not a perfect square, so its square root is irrational and approximately 12.409674.
What is the square root of 154 in simplest radical form?
√154 = √154 in simplest radical form.