Square Root Calculator › Square root of 488
Square Root of 488
√488 ≈ 22.090722
488 is not a perfect square, so √488 is an irrational number that is approximately 22.090722. In simplified radical form, √488 = 2√122.
√488 lies between √484 = 22 and √529 = 23.
| Number | 488 |
| Square root (decimal) | 22.090722 |
| Simplified radical | 2√122 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 488
488 = 2^3 × 61. This is what determines the simplified radical: for √488 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √488 = 2√122.
Estimating √488 by hand
√488 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 488 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.090909 |
| Step 2 | 22.090722 |
| Step 3 | 22.090722 |
After 3 steps the estimate is 22.090722, matching √488 ≈ 22.090722.
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Frequently asked questions
Is 488 a perfect square?
No, 488 is not a perfect square, so its square root is irrational and approximately 22.090722.
What is the square root of 488 in simplest radical form?
√488 = 2√122 in simplest radical form.