Square Root Calculator › Square root of 473
Square Root of 473
√473 ≈ 21.748563
473 is not a perfect square, so √473 is an irrational number that is approximately 21.748563. In simplified radical form, √473 = √473.
√473 lies between √441 = 21 and √484 = 22.
| Number | 473 |
| Square root (decimal) | 21.748563 |
| Simplified radical | √473 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 473
473 = 11 × 43. This is what determines the simplified radical: for √473 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √473 is already in simplest radical form as √473.
Estimating √473 by hand
√473 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 473 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.761905 |
| Step 2 | 21.748567 |
| Step 3 | 21.748563 |
After 3 steps the estimate is 21.748563, matching √473 ≈ 21.748563.
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Frequently asked questions
Is 473 a perfect square?
No, 473 is not a perfect square, so its square root is irrational and approximately 21.748563.
What is the square root of 473 in simplest radical form?
√473 = √473 in simplest radical form.