Square Root Calculator › Square root of 192
Square Root of 192
√192 ≈ 13.856406
192 is not a perfect square, so √192 is an irrational number that is approximately 13.856406. In simplified radical form, √192 = 8√3.
√192 lies between √169 = 13 and √196 = 14.
| Number | 192 |
| Square root (decimal) | 13.856406 |
| Simplified radical | 8√3 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 192
192 = 2^6 × 3. This is what determines the simplified radical: for √192 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √192 = 8√3.
Estimating √192 by hand
√192 falls between √169 = 13 and √196 = 14, so it starts at 13.something. The Babylonian method refines that guess by averaging it with 192 ÷ the guess:
| Step | Estimate |
| Start | 13 |
| Step 1 | 13.884615 |
| Step 2 | 13.856435 |
| Step 3 | 13.856406 |
After 3 steps the estimate is 13.856406, matching √192 ≈ 13.856406.
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Frequently asked questions
Is 192 a perfect square?
No, 192 is not a perfect square, so its square root is irrational and approximately 13.856406.
What is the square root of 192 in simplest radical form?
√192 = 8√3 in simplest radical form.