Square Root Calculator › Square root of 153
Square Root of 153
√153 ≈ 12.369317
153 is not a perfect square, so √153 is an irrational number that is approximately 12.369317. In simplified radical form, √153 = 3√17.
√153 lies between √144 = 12 and √169 = 13.
| Number | 153 |
| Square root (decimal) | 12.369317 |
| Simplified radical | 3√17 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 153
153 = 3^2 × 17. This is what determines the simplified radical: for √153 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √153 = 3√17.
Estimating √153 by hand
√153 falls between √144 = 12 and √169 = 13, so it starts at 12.something. The Babylonian method refines that guess by averaging it with 153 ÷ the guess:
| Step | Estimate |
| Start | 12 |
| Step 1 | 12.375 |
| Step 2 | 12.369318 |
| Step 3 | 12.369317 |
After 3 steps the estimate is 12.369317, matching √153 ≈ 12.369317.
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Frequently asked questions
Is 153 a perfect square?
No, 153 is not a perfect square, so its square root is irrational and approximately 12.369317.
What is the square root of 153 in simplest radical form?
√153 = 3√17 in simplest radical form.