Square Root Calculator › Square root of 45
Square Root of 45
√45 ≈ 6.708204
45 is not a perfect square, so √45 is an irrational number that is approximately 6.708204. In simplified radical form, √45 = 3√5.
√45 lies between √36 = 6 and √49 = 7.
| Number | 45 |
| Square root (decimal) | 6.708204 |
| Simplified radical | 3√5 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 45
45 = 3^2 × 5. This is what determines the simplified radical: for √45 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √45 = 3√5.
Estimating √45 by hand
√45 falls between √36 = 6 and √49 = 7, so it starts at 6.something. The Babylonian method refines that guess by averaging it with 45 ÷ the guess:
| Step | Estimate |
| Start | 6 |
| Step 1 | 6.75 |
| Step 2 | 6.708333 |
| Step 3 | 6.708204 |
After 3 steps the estimate is 6.708204, matching √45 ≈ 6.708204.
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Frequently asked questions
Is 45 a perfect square?
No, 45 is not a perfect square, so its square root is irrational and approximately 6.708204.
What is the square root of 45 in simplest radical form?
√45 = 3√5 in simplest radical form.