Square Root Calculator › Square root of 89
Square Root of 89
√89 ≈ 9.433981
89 is not a perfect square, so √89 is an irrational number that is approximately 9.433981. In simplified radical form, √89 = √89.
√89 lies between √81 = 9 and √100 = 10.
| Number | 89 |
| Square root (decimal) | 9.433981 |
| Simplified radical | √89 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 89
89 = 89. This is what determines the simplified radical: for √89 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √89 is already in simplest radical form as √89.
Estimating √89 by hand
√89 falls between √81 = 9 and √100 = 10, so it starts at 9.something. The Babylonian method refines that guess by averaging it with 89 ÷ the guess:
| Step | Estimate |
| Start | 9 |
| Step 1 | 9.444444 |
| Step 2 | 9.433987 |
| Step 3 | 9.433981 |
After 3 steps the estimate is 9.433981, matching √89 ≈ 9.433981.
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Frequently asked questions
Is 89 a perfect square?
No, 89 is not a perfect square, so its square root is irrational and approximately 9.433981.
What is the square root of 89 in simplest radical form?
√89 = √89 in simplest radical form.