√95 at a glance
- Exact value
- √95
- Decimal (10 places)
- 9.7467943448
- Rounded
- 9.7 · 9.75 · 9.747
- Perfect square?
- No — between 9² and 10²
- Rational?
- Irrational
- Both square roots
- ±9.746794
- Prime factorization
- 5 × 19
- Cube root
- 4.562903
How to simplify √95
The prime factorization of 95 is 5 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √95 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 95, 5 and 19 appear an odd number of times, so √95 is irrational and 9.7467943448 is a rounded value.
Where √95 sits between perfect squares
81 = 9² and 100 = 10² are the nearest perfect squares, so √95 lies between 9 and 10. 95 is 14 above 81 and 5 below 100, so the root is closer to 10.
- Straight line between 81 and 100: 9.7368 (0.1% low)
- Tangent from 9, i.e. 9 + 14 ÷ 18: 9.7778 (0.32% high)
- Tangent from 10, i.e. 10 − 5 ÷ 20: 9.7500 (0.03% high)
For √95 the tangent at 10 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 95 is just 5 below 100.
Finding √95 with the Babylonian method
If a guess is too big, 95 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√95) in one step.
Start from the nearest whole number, 10 (10² = 100):
| Step | Guess x | 95 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 9.5000000000 | 9.7500000000 | 2 |
| 2 | 9.7500000000 | 9.7435897436 | 9.7467948718 | 6 |
| 3 | 9.7467948718 | 9.7467938178 | 9.7467943448 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √95 = 9.7467943448 to every decimal shown.
√95 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √95 the pattern is [9; 1, 2, 1, 18] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √95 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 9/1 | 9.0000000000 | 7.5 × 10⁻¹ |
| 10/1 | 10.0000000000 | 2.5 × 10⁻¹ |
| 29/3 | 9.6666666667 | 8.0 × 10⁻² |
| 39/4 | 9.7500000000 | 3.2 × 10⁻³ |
| 731/75 | 9.7466666667 | 1.3 × 10⁻⁴ |
| 770/79 | 9.7468354430 | 4.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 95y² = 1. Its smallest solution in positive whole numbers is x = 39, y = 4.
√95 in geometry and everyday measurements
- A square room or garden bed covering 95 square feet measures about 9.75 ft (9 ft 9 in) along each wall.
- 95 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √95 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √95 as its space diagonal.
Square roots near √95 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √92 | 2√23 | 9.5917 | No |
| √93 | √93 | 9.6437 | No |
| √94 | √94 | 9.6954 | No |
| √95 | √95 | 9.7468 | No |
| √96 | 4√6 | 9.7980 | No |
| √97 | √97 | 9.8489 | No |
| √98 | 7√2 | 9.8995 | No |
- The cube root of 95 is about 4.562903.
- Four times the radicand doubles the root: √380 = 2 × √95 ≈ 19.493589.
Frequently asked questions
What is the square root of 95?
The square root of 95 is √95, about 9.7467943448. The negative root, −9.746794, also squares to 95.
Is the square root of 95 rational or irrational?
Irrational. 95 is not a perfect square — it falls between 81 and 100 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √95 be simplified?
No. 95 = 5 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √95 rounded to two decimal places?
√95 ≈ 9.75 to two decimal places (9.7 to one, 9.747 to three). Check: 9.75² = 95.0625, close to 95.