Square Root Calculator › Square root of 165
Square Root of 165
√165 ≈ 12.845233
165 is not a perfect square, so √165 is an irrational number that is approximately 12.845233. In simplified radical form, √165 = √165.
√165 lies between √144 = 12 and √169 = 13.
| Number | 165 |
| Square root (decimal) | 12.845233 |
| Simplified radical | √165 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 165
165 = 3 × 5 × 11. This is what determines the simplified radical: for √165 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √165 is already in simplest radical form as √165.
Estimating √165 by hand
√165 falls between √144 = 12 and √169 = 13, so it starts at 12.something. The Babylonian method refines that guess by averaging it with 165 ÷ the guess:
| Step | Estimate |
| Start | 12 |
| Step 1 | 12.875 |
| Step 2 | 12.845267 |
| Step 3 | 12.845233 |
After 3 steps the estimate is 12.845233, matching √165 ≈ 12.845233.
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Frequently asked questions
Is 165 a perfect square?
No, 165 is not a perfect square, so its square root is irrational and approximately 12.845233.
What is the square root of 165 in simplest radical form?
√165 = √165 in simplest radical form.