Square Root Calculator › Square root of 457
Square Root of 457
√457 ≈ 21.377558
457 is not a perfect square, so √457 is an irrational number that is approximately 21.377558. In simplified radical form, √457 = √457.
√457 lies between √441 = 21 and √484 = 22.
| Number | 457 |
| Square root (decimal) | 21.377558 |
| Simplified radical | √457 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 457
457 = 457. This is what determines the simplified radical: for √457 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √457 is already in simplest radical form as √457.
Estimating √457 by hand
√457 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 457 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.380952 |
| Step 2 | 21.377559 |
| Step 3 | 21.377558 |
After 3 steps the estimate is 21.377558, matching √457 ≈ 21.377558.
Nearby square roots
Frequently asked questions
Is 457 a perfect square?
No, 457 is not a perfect square, so its square root is irrational and approximately 21.377558.
What is the square root of 457 in simplest radical form?
√457 = √457 in simplest radical form.