Square Root Calculator › Square root of 474
Square Root of 474
√474 ≈ 21.771541
474 is not a perfect square, so √474 is an irrational number that is approximately 21.771541. In simplified radical form, √474 = √474.
√474 lies between √441 = 21 and √484 = 22.
| Number | 474 |
| Square root (decimal) | 21.771541 |
| Simplified radical | √474 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 474
474 = 2 × 3 × 79. This is what determines the simplified radical: for √474 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √474 is already in simplest radical form as √474.
Estimating √474 by hand
√474 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 474 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.785714 |
| Step 2 | 21.771546 |
| Step 3 | 21.771541 |
After 3 steps the estimate is 21.771541, matching √474 ≈ 21.771541.
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Frequently asked questions
Is 474 a perfect square?
No, 474 is not a perfect square, so its square root is irrational and approximately 21.771541.
What is the square root of 474 in simplest radical form?
√474 = √474 in simplest radical form.