Square Root Calculator › Square root of 306
Square Root of 306
√306 ≈ 17.492856
306 is not a perfect square, so √306 is an irrational number that is approximately 17.492856. In simplified radical form, √306 = 3√34.
√306 lies between √289 = 17 and √324 = 18.
| Number | 306 |
| Square root (decimal) | 17.492856 |
| Simplified radical | 3√34 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 306
306 = 2 × 3^2 × 17. This is what determines the simplified radical: for √306 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √306 = 3√34.
Estimating √306 by hand
√306 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 306 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.5 |
| Step 2 | 17.492857 |
| Step 3 | 17.492856 |
After 3 steps the estimate is 17.492856, matching √306 ≈ 17.492856.
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Frequently asked questions
Is 306 a perfect square?
No, 306 is not a perfect square, so its square root is irrational and approximately 17.492856.
What is the square root of 306 in simplest radical form?
√306 = 3√34 in simplest radical form.