Square Root Calculator › Square root of 189
Square Root of 189
√189 ≈ 13.747727
189 is not a perfect square, so √189 is an irrational number that is approximately 13.747727. In simplified radical form, √189 = 3√21.
√189 lies between √169 = 13 and √196 = 14.
| Number | 189 |
| Square root (decimal) | 13.747727 |
| Simplified radical | 3√21 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 189
189 = 3^3 × 7. This is what determines the simplified radical: for √189 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √189 = 3√21.
Estimating √189 by hand
√189 falls between √169 = 13 and √196 = 14, so it starts at 13.something. The Babylonian method refines that guess by averaging it with 189 ÷ the guess:
| Step | Estimate |
| Start | 13 |
| Step 1 | 13.769231 |
| Step 2 | 13.747744 |
| Step 3 | 13.747727 |
After 3 steps the estimate is 13.747727, matching √189 ≈ 13.747727.
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Frequently asked questions
Is 189 a perfect square?
No, 189 is not a perfect square, so its square root is irrational and approximately 13.747727.
What is the square root of 189 in simplest radical form?
√189 = 3√21 in simplest radical form.