Square Root Calculator › Square root of 465
Square Root of 465
√465 ≈ 21.563859
465 is not a perfect square, so √465 is an irrational number that is approximately 21.563859. In simplified radical form, √465 = √465.
√465 lies between √441 = 21 and √484 = 22.
| Number | 465 |
| Square root (decimal) | 21.563859 |
| Simplified radical | √465 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 465
465 = 3 × 5 × 31. This is what determines the simplified radical: for √465 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √465 is already in simplest radical form as √465.
Estimating √465 by hand
√465 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 465 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.571429 |
| Step 2 | 21.56386 |
| Step 3 | 21.563859 |
After 3 steps the estimate is 21.563859, matching √465 ≈ 21.563859.
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Frequently asked questions
Is 465 a perfect square?
No, 465 is not a perfect square, so its square root is irrational and approximately 21.563859.
What is the square root of 465 in simplest radical form?
√465 = √465 in simplest radical form.