Square Root Calculator › Square root of 469
Square Root of 469
√469 ≈ 21.656408
469 is not a perfect square, so √469 is an irrational number that is approximately 21.656408. In simplified radical form, √469 = √469.
√469 lies between √441 = 21 and √484 = 22.
| Number | 469 |
| Square root (decimal) | 21.656408 |
| Simplified radical | √469 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 469
469 = 7 × 67. This is what determines the simplified radical: for √469 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √469 is already in simplest radical form as √469.
Estimating √469 by hand
√469 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 469 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.666667 |
| Step 2 | 21.65641 |
| Step 3 | 21.656408 |
After 3 steps the estimate is 21.656408, matching √469 ≈ 21.656408.
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Frequently asked questions
Is 469 a perfect square?
No, 469 is not a perfect square, so its square root is irrational and approximately 21.656408.
What is the square root of 469 in simplest radical form?
√469 = √469 in simplest radical form.