√832 at a glance
- Exact value
- 8√13
- Decimal (10 places)
- 28.8444102037
- Rounded
- 28.8 · 28.84 · 28.844
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.844410
- Prime factorization
- 2⁶ × 13
- Cube root
- 9.405339
How to simplify √832
Look for the largest perfect square that divides 832. Here it is 64 (8²), because 832 = 64 × 13 and 13 has no square factor left:
The prime factorization tells the same story: 832 = 2⁶ × 13. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 13 stays inside.
832 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √832 = 2√208, and √208 can be simplified again. Using 64 straight away finishes in one step.
Check: (8√13)² = 8² × 13 = 64 × 13 = 832. As a decimal, 8√13 = 8 × 3.6055512755 ≈ 28.8444102037.
Where √832 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √832 lies between 28 and 29. 832 is 48 above 784 and 9 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.8421 (0.01% low)
- Tangent from 28, i.e. 28 + 48 ÷ 56: 28.8571 (0.04% high)
- Tangent from 29, i.e. 29 − 9 ÷ 58: 28.8448 (0% high)
For √832 the tangent at 29 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 832 is just 9 below 841.
Finding √832 with the Babylonian method
Picture a rectangle with an area of 832 and one side x; the other side must be 832 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √832.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 832 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.6896551724 | 28.8448275862 | 3 |
| 2 | 28.8448275862 | 28.8439928273 | 28.8444102067 | 8 |
| 3 | 28.8444102067 | 28.8444102007 | 28.8444102037 | 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 √832 = 28.8444102037 to every decimal shown.
√832 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √832 the pattern is [28; 1, 5, 2, 2, 1, 13, 1, 2, 2, 5, 1, 56] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √832 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 | 8.4 × 10⁻¹ |
| 29/1 | 29.0000000000 | 1.6 × 10⁻¹ |
| 173/6 | 28.8333333333 | 1.1 × 10⁻² |
| 375/13 | 28.8461538462 | 1.7 × 10⁻³ |
| 923/32 | 28.8437500000 | 6.6 × 10⁻⁴ |
| 1,298/45 | 28.8444444444 | 3.4 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 832y² = 1. Its smallest solution in positive whole numbers is x = 842,401, y = 29,205.
√832 in geometry and everyday measurements
- 832 square feet is 77.3 m². Laid out as a square — a small house footprint or a lot — it is about 28.84 ft (28 ft 10 in) on a side.
- 832 = 16² + 24², so by the Pythagorean theorem √832 is the diagonal of a 16 × 24 rectangle — and the distance between the points (0, 0) and (16, 24) on a grid.
- Since √832 = 8√13, a length of √832 is exactly 8 copies of the length √13 laid end to end.
Square roots near √832 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √829 | √829 | 28.7924 | No |
| √830 | √830 | 28.8097 | No |
| √831 | √831 | 28.8271 | No |
| √832 | 8√13 | 28.8444 | No |
| √833 | 7√17 | 28.8617 | No |
| √834 | √834 | 28.8791 | No |
| √835 | √835 | 28.8964 | No |
- The cube root of 832 is about 9.405339.
- Because 832 = 4 × 208, the root is twice √208: 2 × 14.422205 ≈ 28.84441.
Frequently asked questions
What is the square root of 832?
The square root of 832 is 8√13 in simplest radical form, which is about 28.8444102037. The negative root, −28.844410, also squares to 832.
Is the square root of 832 rational or irrational?
Irrational. 832 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 √832 be simplified?
Yes. The largest perfect square dividing 832 is 64, so √832 = √64 × √13 = 8√13.
What is √832 rounded to two decimal places?
√832 ≈ 28.84 to two decimal places (28.8 to one, 28.844 to three). Check: 28.84² = 831.7456, close to 832.