Square Root Calculator › Square root of 420
Square Root of 420
√420 ≈ 20.493902
420 is not a perfect square, so √420 is an irrational number that is approximately 20.493902. In simplified radical form, √420 = 2√105.
√420 lies between √400 = 20 and √441 = 21.
| Number | 420 |
| Square root (decimal) | 20.493902 |
| Simplified radical | 2√105 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 420
420 = 2^2 × 3 × 5 × 7. This is what determines the simplified radical: for √420 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √420 = 2√105.
Estimating √420 by hand
√420 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 420 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.5 |
| Step 2 | 20.493902 |
| Step 3 | 20.493902 |
After 3 steps the estimate is 20.493902, matching √420 ≈ 20.493902.
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Frequently asked questions
Is 420 a perfect square?
No, 420 is not a perfect square, so its square root is irrational and approximately 20.493902.
What is the square root of 420 in simplest radical form?
√420 = 2√105 in simplest radical form.