Square Root Calculator › Square root of 52
Square Root of 52
√52 ≈ 7.211103
52 is not a perfect square, so √52 is an irrational number that is approximately 7.211103. In simplified radical form, √52 = 2√13.
√52 lies between √49 = 7 and √64 = 8.
| Number | 52 |
| Square root (decimal) | 7.211103 |
| Simplified radical | 2√13 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 52
52 = 2^2 × 13. This is what determines the simplified radical: for √52 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √52 = 2√13.
Estimating √52 by hand
√52 falls between √49 = 7 and √64 = 8, so it starts at 7.something. The Babylonian method refines that guess by averaging it with 52 ÷ the guess:
| Step | Estimate |
| Start | 7 |
| Step 1 | 7.214286 |
| Step 2 | 7.211103 |
| Step 3 | 7.211103 |
After 3 steps the estimate is 7.211103, matching √52 ≈ 7.211103.
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Frequently asked questions
Is 52 a perfect square?
No, 52 is not a perfect square, so its square root is irrational and approximately 7.211103.
What is the square root of 52 in simplest radical form?
√52 = 2√13 in simplest radical form.