Square Root of 95
The square root of 95 is approximately 9.746794, or exactly √95 in simplified radical form. No, 95 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √95 sits
9² = 81 and 10² = 100, and 95 falls between them, so √95 is between 9 and 10. It's closer to 10 (14 above 81, 5 below 100), which matches the decimal 9.746794.
| k | k² | Compared to 95 |
|---|---|---|
| 7 | 49 | 46 below |
| 8 | 64 | 31 below |
| 9 | 81 | 14 below |
| 10 | 100 | 5 above |
| 11 | 121 | 26 above |
| 12 | 144 | 49 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 95 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 95 ÷ x | New guess (x + 95÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.555556 | 9.777778 |
| 2 | 9.777778 | 9.715909 | 9.746843 |
| 3 | 9.746843 | 9.746745 | 9.746794 |
| 4 | 9.746794 | 9.746794 | 9.746794 |
After 4 passes the guess is 9.746794, already matching √95 = 9.746794 to six decimal places.
Exact form from the prime factors
95 = 5 × 19. No prime appears twice, so nothing can be pulled out of the root -- √95 is already in simplest form.
Frequently Asked Questions
What is the square root of 95?
The square root of 95 is approximately 9.746794, or exactly √95 in simplified radical form.
Is 95 a perfect square?
No, 95 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √95 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 95?
√94 is about 9.6954, and √96 is about 9.798. 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 √95 squared?
Exactly 95 -- squaring undoes the square root. Squaring the rounded decimal 9.746794 gives 95, which is why the radical form √95 is the exact answer and the decimal is an approximation.
What is the negative square root of 95?
−9.746794. Every positive number has two square roots, one positive and one negative, because (−9.746794)² is also 95. The radical sign √ by convention means the positive one.
More Number Theory
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