Square Root Calculator › Square root of 403
Square Root of 403
√403 ≈ 20.07486
403 is not a perfect square, so √403 is an irrational number that is approximately 20.07486. In simplified radical form, √403 = √403.
√403 lies between √400 = 20 and √441 = 21.
| Number | 403 |
| Square root (decimal) | 20.07486 |
| Simplified radical | √403 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 403
403 = 13 × 31. This is what determines the simplified radical: for √403 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √403 is already in simplest radical form as √403.
Estimating √403 by hand
√403 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 403 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.075 |
| Step 2 | 20.07486 |
| Step 3 | 20.07486 |
After 3 steps the estimate is 20.07486, matching √403 ≈ 20.07486.
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Frequently asked questions
Is 403 a perfect square?
No, 403 is not a perfect square, so its square root is irrational and approximately 20.07486.
What is the square root of 403 in simplest radical form?
√403 = √403 in simplest radical form.