Square Root Calculator › Square root of 21
Square Root of 21
√21 ≈ 4.582576
21 is not a perfect square, so √21 is an irrational number that is approximately 4.582576. In simplified radical form, √21 = √21.
√21 lies between √16 = 4 and √25 = 5.
| Number | 21 |
| Square root (decimal) | 4.582576 |
| Simplified radical | √21 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 21
21 = 3 × 7. This is what determines the simplified radical: for √21 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √21 is already in simplest radical form as √21.
Estimating √21 by hand
√21 falls between √16 = 4 and √25 = 5, so it starts at 4.something. The Babylonian method refines that guess by averaging it with 21 ÷ the guess:
| Step | Estimate |
| Start | 4 |
| Step 1 | 4.625 |
| Step 2 | 4.58277 |
| Step 3 | 4.582576 |
After 3 steps the estimate is 4.582576, matching √21 ≈ 4.582576.
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Frequently asked questions
Is 21 a perfect square?
No, 21 is not a perfect square, so its square root is irrational and approximately 4.582576.
What is the square root of 21 in simplest radical form?
√21 = √21 in simplest radical form.