√795 at a glance
- Exact value
- √795
- Decimal (10 places)
- 28.1957443597
- Rounded
- 28.2 · 28.20 · 28.196
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.195744
- Prime factorization
- 3 × 5 × 53
- Cube root
- 9.263797
How to simplify √795
The prime factorization of 795 is 3 × 5 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √795 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 795, 3, 5 and 53 appear an odd number of times, so √795 is irrational and 28.1957443597 is a rounded value.
Where √795 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √795 lies between 28 and 29. 795 is 11 above 784 and 46 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.1930 (0.01% low)
- Tangent from 28, i.e. 28 + 11 ÷ 56: 28.1964 (0% high)
- Tangent from 29, i.e. 29 − 46 ÷ 58: 28.2069 (0.04% high)
For √795 the tangent at 28 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 795 is just 11 above 784.
Finding √795 with the Babylonian method
If a guess is too big, 795 ÷ 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√795) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 795 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.3928571429 | 28.1964285714 | 3 |
| 2 | 28.1964285714 | 28.1950601647 | 28.1957443680 | 8 |
| 3 | 28.1957443680 | 28.1957443514 | 28.1957443597 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √795 = 28.1957443597 to every decimal shown.
√795 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √795 the pattern is [28; 5, 9, 5, 56] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √795 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 2.0 × 10⁻¹ |
| 141/5 | 28.2000000000 | 4.3 × 10⁻³ |
| 1,297/46 | 28.1956521739 | 9.2 × 10⁻⁵ |
| 6,626/235 | 28.1957446809 | 3.2 × 10⁻⁷ |
| 372,353/13,206 | 28.1957443586 | 1.1 × 10⁻⁹ |
| 1,868,391/66,265 | 28.1957443598 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 795y² = 1. Its smallest solution in positive whole numbers is x = 6,626, y = 235.
√795 in geometry and everyday measurements
- 795 square feet is 73.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.2 ft (28 ft 2 in) on a side.
- 795 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √795 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 13 × 25 box, because 1² + 13² + 25² = 795.
Square roots near √795 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √792 | 6√22 | 28.1425 | No |
| √793 | √793 | 28.1603 | No |
| √794 | √794 | 28.1780 | No |
| √795 | √795 | 28.1957 | No |
| √796 | 2√199 | 28.2135 | No |
| √797 | √797 | 28.2312 | No |
| √798 | √798 | 28.2489 | No |
- The cube root of 795 is about 9.263797.
- Squaring undoes the root: (√795)² = 795, while 795² = 632,025 — the number whose square root is 795.
Frequently asked questions
What is the square root of 795?
The square root of 795 is √795, about 28.1957443597. The negative root, −28.195744, also squares to 795.
Is the square root of 795 rational or irrational?
Irrational. 795 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √795 be simplified?
No. 795 = 3 × 5 × 53 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √795 rounded to two decimal places?
√795 ≈ 28.20 to two decimal places (28.2 to one, 28.196 to three). Check: 28.20² = 795.24, close to 795.