√835 at a glance
- Exact value
- √835
- Decimal (10 places)
- 28.8963665536
- Rounded
- 28.9 · 28.90 · 28.896
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.896367
- Prime factorization
- 5 × 167
- Cube root
- 9.416630
How to simplify √835
The prime factorization of 835 is 5 × 167. Every prime appears only once, so there is no pair to bring outside the radical — √835 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 835, 5 and 167 appear an odd number of times, so √835 is irrational and 28.8963665536 is a rounded value.
Where √835 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √835 lies between 28 and 29. 835 is 51 above 784 and 6 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.8947 (0.01% low)
- Tangent from 28, i.e. 28 + 51 ÷ 56: 28.9107 (0.05% high)
- Tangent from 29, i.e. 29 − 6 ÷ 58: 28.8966 (0% high)
For √835 the tangent at 29 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 835 is just 6 below 841.
Finding √835 with the Babylonian method
If a guess is too big, 835 ÷ 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√835) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 835 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.7931034483 | 28.8965517241 | 3 |
| 2 | 28.8965517241 | 28.8961813842 | 28.8963665542 | 9 |
| 3 | 28.8963665542 | 28.8963665530 | 28.8963665536 | 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 √835 = 28.8963665536 to every decimal shown.
√835 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √835 the pattern is [28; 1, 8, 1, 1, 1, 5, 1, 3, 3, 1, 1, 2, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √835 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 | 9.0 × 10⁻¹ |
| 29/1 | 29.0000000000 | 1.0 × 10⁻¹ |
| 260/9 | 28.8888888889 | 7.5 × 10⁻³ |
| 289/10 | 28.9000000000 | 3.6 × 10⁻³ |
| 549/19 | 28.8947368421 | 1.6 × 10⁻³ |
| 838/29 | 28.8965517241 | 1.9 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 835y² = 1. Its smallest solution in positive whole numbers is x = 34,336,355,806, y = 1,188,258,591.
√835 in geometry and everyday measurements
- 835 square feet is 77.6 m². Laid out as a square — a small house footprint or a lot — it is about 28.9 ft (28 ft 11 in) on a side.
- 835 is not a sum of two whole-number squares — the prime factor 167 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √835 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 9 × 27 box, because 5² + 9² + 27² = 835.
Square roots near √835 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √832 | 8√13 | 28.8444 | No |
| √833 | 7√17 | 28.8617 | No |
| √834 | √834 | 28.8791 | No |
| √835 | √835 | 28.8964 | No |
| √836 | 2√209 | 28.9137 | No |
| √837 | 3√93 | 28.9310 | No |
| √838 | √838 | 28.9482 | No |
- The cube root of 835 is about 9.416630.
- Squaring undoes the root: (√835)² = 835, while 835² = 697,225 — the number whose square root is 835.
Frequently asked questions
What is the square root of 835?
The square root of 835 is √835, about 28.8963665536. The negative root, −28.896367, also squares to 835.
Is the square root of 835 rational or irrational?
Irrational. 835 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 √835 be simplified?
No. 835 = 5 × 167 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √835 rounded to two decimal places?
√835 ≈ 28.90 to two decimal places (28.9 to one, 28.896 to three). Check: 28.90² = 835.21, close to 835.