Square Root Calculator › Square root of 75
Square Root of 75
√75 ≈ 8.660254
75 is not a perfect square, so √75 is an irrational number that is approximately 8.660254. In simplified radical form, √75 = 5√3.
√75 lies between √64 = 8 and √81 = 9.
| Number | 75 |
| Square root (decimal) | 8.660254 |
| Simplified radical | 5√3 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 75
75 = 3 × 5^2. This is what determines the simplified radical: for √75 you pull out one copy of any prime that appears twice.
Here the paired factor 5 comes out from under the root, leaving √75 = 5√3.
Estimating √75 by hand
√75 falls between √64 = 8 and √81 = 9, so it starts at 8.something. The Babylonian method refines that guess by averaging it with 75 ÷ the guess:
| Step | Estimate |
| Start | 8 |
| Step 1 | 8.6875 |
| Step 2 | 8.660297 |
| Step 3 | 8.660254 |
After 3 steps the estimate is 8.660254, matching √75 ≈ 8.660254.
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Frequently asked questions
Is 75 a perfect square?
No, 75 is not a perfect square, so its square root is irrational and approximately 8.660254.
What is the square root of 75 in simplest radical form?
√75 = 5√3 in simplest radical form.