Square Root of 75
The square root of 75 is approximately 8.660254, or exactly 5√3 in simplified radical form. No, 75 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √75 sits
8² = 64 and 9² = 81, and 75 falls between them, so √75 is between 8 and 9. It's closer to 9 (11 above 64, 6 below 81), which matches the decimal 8.660254.
| k | k² | Compared to 75 |
|---|---|---|
| 6 | 36 | 39 below |
| 7 | 49 | 26 below |
| 8 | 64 | 11 below |
| 9 | 81 | 6 above |
| 10 | 100 | 25 above |
| 11 | 121 | 46 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 75 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 75 ÷ x | New guess (x + 75÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9.375 | 8.6875 |
| 2 | 8.6875 | 8.633094 | 8.660297 |
| 3 | 8.660297 | 8.660211 | 8.660254 |
| 4 | 8.660254 | 8.660254 | 8.660254 |
After 4 passes the guess is 8.660254, already matching √75 = 8.660254 to six decimal places.
Exact form from the prime factors
75 = 3 × 5^2. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √75 = 5√3. That's the exact value; the decimal 8.660254 is a rounding of it.
Frequently Asked Questions
What is the square root of 75?
The square root of 75 is approximately 8.660254, or exactly 5√3 in simplified radical form.
Is 75 a perfect square?
No, 75 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √75 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 75?
√74 is about 8.6023, and √76 is about 8.7178. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √75 squared?
Exactly 75 -- squaring undoes the square root. Squaring the rounded decimal 8.660254 gives 75, which is why the radical form 5√3 is the exact answer and the decimal is an approximation.
What is the negative square root of 75?
−8.660254. Every positive number has two square roots, one positive and one negative, because (−8.660254)² is also 75. The radical sign √ by convention means the positive one.
More Number Theory
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