Square Root Calculator › Square root of 133
Square Root of 133
√133 ≈ 11.532563
133 is not a perfect square, so √133 is an irrational number that is approximately 11.532563. In simplified radical form, √133 = √133.
√133 lies between √121 = 11 and √144 = 12.
| Number | 133 |
| Square root (decimal) | 11.532563 |
| Simplified radical | √133 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 133
133 = 7 × 19. This is what determines the simplified radical: for √133 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √133 is already in simplest radical form as √133.
Estimating √133 by hand
√133 falls between √121 = 11 and √144 = 12, so it starts at 11.something. The Babylonian method refines that guess by averaging it with 133 ÷ the guess:
| Step | Estimate |
| Start | 11 |
| Step 1 | 11.545455 |
| Step 2 | 11.53257 |
| Step 3 | 11.532563 |
After 3 steps the estimate is 11.532563, matching √133 ≈ 11.532563.
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Frequently asked questions
Is 133 a perfect square?
No, 133 is not a perfect square, so its square root is irrational and approximately 11.532563.
What is the square root of 133 in simplest radical form?
√133 = √133 in simplest radical form.