√895 at a glance
- Exact value
- √895
- Decimal (10 places)
- 29.9165506033
- Rounded
- 29.9 · 29.92 · 29.917
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.916551
- Prime factorization
- 5 × 179
- Cube root
- 9.636981
How to simplify √895
The prime factorization of 895 is 5 × 179. Every prime appears only once, so there is no pair to bring outside the radical — √895 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 895, 5 and 179 appear an odd number of times, so √895 is irrational and 29.9165506033 is a rounded value.
Where √895 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √895 lies between 29 and 30. 895 is 54 above 841 and 5 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.9153 (0% low)
- Tangent from 29, i.e. 29 + 54 ÷ 58: 29.9310 (0.05% high)
- Tangent from 30, i.e. 30 − 5 ÷ 60: 29.9167 (0% high)
For √895 the tangent at 30 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 895 is just 5 below 900.
Finding √895 with the Babylonian method
If a guess is too big, 895 ÷ 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√895) in one step.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 895 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.8333333333 | 29.9166666667 | 3 |
| 2 | 29.9166666667 | 29.9164345404 | 29.9165506035 | 9 |
| 3 | 29.9165506035 | 29.9165506031 | 29.9165506033 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √895 = 29.9165506033 to every decimal shown.
√895 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √895 the pattern is [29; 1, 10, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √895 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 9.2 × 10⁻¹ |
| 30/1 | 30.0000000000 | 8.3 × 10⁻² |
| 329/11 | 29.9090909091 | 7.5 × 10⁻³ |
| 359/12 | 29.9166666667 | 1.2 × 10⁻⁴ |
| 21,151/707 | 29.9165487977 | 1.8 × 10⁻⁶ |
| 21,510/719 | 29.9165507650 | 1.6 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 895y² = 1. Its smallest solution in positive whole numbers is x = 359, y = 12.
√895 in geometry and everyday measurements
- 895 square feet is 83.1 m². Laid out as a square — a small house footprint or a lot — it is about 29.92 ft (29 ft 11 in) on a side.
- 895 is not a sum of two whole-number squares — the prime factor 179 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √895 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 √895 as its space diagonal.
Square roots near √895 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √892 | 2√223 | 29.8664 | No |
| √893 | √893 | 29.8831 | No |
| √894 | √894 | 29.8998 | No |
| √895 | √895 | 29.9166 | No |
| √896 | 8√14 | 29.9333 | No |
| √897 | √897 | 29.9500 | No |
| √898 | √898 | 29.9666 | No |
- The cube root of 895 is about 9.636981.
- Squaring undoes the root: (√895)² = 895, while 895² = 801,025 — the number whose square root is 895.
Frequently asked questions
What is the square root of 895?
The square root of 895 is √895, about 29.9165506033. The negative root, −29.916551, also squares to 895.
Is the square root of 895 rational or irrational?
Irrational. 895 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √895 be simplified?
No. 895 = 5 × 179 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √895 rounded to two decimal places?
√895 ≈ 29.92 to two decimal places (29.9 to one, 29.917 to three). Check: 29.92² = 895.2064, close to 895.