Square Root Calculator › Square root of 402
Square Root of 402
√402 ≈ 20.049938
402 is not a perfect square, so √402 is an irrational number that is approximately 20.049938. In simplified radical form, √402 = √402.
√402 lies between √400 = 20 and √441 = 21.
| Number | 402 |
| Square root (decimal) | 20.049938 |
| Simplified radical | √402 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 402
402 = 2 × 3 × 67. This is what determines the simplified radical: for √402 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √402 is already in simplest radical form as √402.
Estimating √402 by hand
√402 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 402 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.05 |
| Step 2 | 20.049938 |
| Step 3 | 20.049938 |
After 3 steps the estimate is 20.049938, matching √402 ≈ 20.049938.
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Frequently asked questions
Is 402 a perfect square?
No, 402 is not a perfect square, so its square root is irrational and approximately 20.049938.
What is the square root of 402 in simplest radical form?
√402 = √402 in simplest radical form.