Square Root Calculator › Square root of 456
Square Root of 456
√456 ≈ 21.354157
456 is not a perfect square, so √456 is an irrational number that is approximately 21.354157. In simplified radical form, √456 = 2√114.
√456 lies between √441 = 21 and √484 = 22.
| Number | 456 |
| Square root (decimal) | 21.354157 |
| Simplified radical | 2√114 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 456
456 = 2^3 × 3 × 19. This is what determines the simplified radical: for √456 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √456 = 2√114.
Estimating √456 by hand
√456 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 456 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.357143 |
| Step 2 | 21.354157 |
| Step 3 | 21.354157 |
After 3 steps the estimate is 21.354157, matching √456 ≈ 21.354157.
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Frequently asked questions
Is 456 a perfect square?
No, 456 is not a perfect square, so its square root is irrational and approximately 21.354157.
What is the square root of 456 in simplest radical form?
√456 = 2√114 in simplest radical form.