Square Root Calculator › Square root of 147
Square Root of 147
√147 ≈ 12.124356
147 is not a perfect square, so √147 is an irrational number that is approximately 12.124356. In simplified radical form, √147 = 7√3.
√147 lies between √144 = 12 and √169 = 13.
| Number | 147 |
| Square root (decimal) | 12.124356 |
| Simplified radical | 7√3 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 147
147 = 3 × 7^2. This is what determines the simplified radical: for √147 you pull out one copy of any prime that appears twice.
Here the paired factor 7 comes out from under the root, leaving √147 = 7√3.
Estimating √147 by hand
√147 falls between √144 = 12 and √169 = 13, so it starts at 12.something. The Babylonian method refines that guess by averaging it with 147 ÷ the guess:
| Step | Estimate |
| Start | 12 |
| Step 1 | 12.125 |
| Step 2 | 12.124356 |
| Step 3 | 12.124356 |
After 3 steps the estimate is 12.124356, matching √147 ≈ 12.124356.
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Frequently asked questions
Is 147 a perfect square?
No, 147 is not a perfect square, so its square root is irrational and approximately 12.124356.
What is the square root of 147 in simplest radical form?
√147 = 7√3 in simplest radical form.