Square Root Calculator › Square root of 345
Square Root of 345
√345 ≈ 18.574176
345 is not a perfect square, so √345 is an irrational number that is approximately 18.574176. In simplified radical form, √345 = √345.
√345 lies between √324 = 18 and √361 = 19.
| Number | 345 |
| Square root (decimal) | 18.574176 |
| Simplified radical | √345 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 345
345 = 3 × 5 × 23. This is what determines the simplified radical: for √345 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √345 is already in simplest radical form as √345.
Estimating √345 by hand
√345 falls between √324 = 18 and √361 = 19, so it starts at 18.something. The Babylonian method refines that guess by averaging it with 345 ÷ the guess:
| Step | Estimate |
| Start | 18 |
| Step 1 | 18.583333 |
| Step 2 | 18.574178 |
| Step 3 | 18.574176 |
After 3 steps the estimate is 18.574176, matching √345 ≈ 18.574176.
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Frequently asked questions
Is 345 a perfect square?
No, 345 is not a perfect square, so its square root is irrational and approximately 18.574176.
What is the square root of 345 in simplest radical form?
√345 = √345 in simplest radical form.