Square Root Calculator › Square root of 208
Square Root of 208
√208 ≈ 14.422205
208 is not a perfect square, so √208 is an irrational number that is approximately 14.422205. In simplified radical form, √208 = 4√13.
√208 lies between √196 = 14 and √225 = 15.
| Number | 208 |
| Square root (decimal) | 14.422205 |
| Simplified radical | 4√13 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 208
208 = 2^4 × 13. This is what determines the simplified radical: for √208 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √208 = 4√13.
Estimating √208 by hand
√208 falls between √196 = 14 and √225 = 15, so it starts at 14.something. The Babylonian method refines that guess by averaging it with 208 ÷ the guess:
| Step | Estimate |
| Start | 14 |
| Step 1 | 14.428571 |
| Step 2 | 14.422207 |
| Step 3 | 14.422205 |
After 3 steps the estimate is 14.422205, matching √208 ≈ 14.422205.
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Frequently asked questions
Is 208 a perfect square?
No, 208 is not a perfect square, so its square root is irrational and approximately 14.422205.
What is the square root of 208 in simplest radical form?
√208 = 4√13 in simplest radical form.