√928 at a glance
- Exact value
- 4√58
- Decimal (10 places)
- 30.4630924235
- Rounded
- 30.5 · 30.46 · 30.463
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.463092
- Prime factorization
- 2⁵ × 29
- Cube root
- 9.753998
How to simplify √928
Look for the largest perfect square that divides 928. Here it is 16 (4²), because 928 = 16 × 58 and 58 has no square factor left:
The prime factorization tells the same story: 928 = 2⁵ × 29. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 29 stays inside.
928 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √928 = 2√232, and √232 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√58)² = 4² × 58 = 16 × 58 = 928. As a decimal, 4√58 = 4 × 7.6157731059 ≈ 30.4630924235.
Where √928 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √928 lies between 30 and 31. 928 is 28 above 900 and 33 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.4590 (0.01% low)
- Tangent from 30, i.e. 30 + 28 ÷ 60: 30.4667 (0.01% high)
- Tangent from 31, i.e. 31 − 33 ÷ 62: 30.4677 (0.02% high)
For √928 the tangent at 30 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 928 is just 28 above 900.
Finding √928 with the Babylonian method
Picture a rectangle with an area of 928 and one side x; the other side must be 928 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √928.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 928 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.9333333333 | 30.4666666667 | 2 |
| 2 | 30.4666666667 | 30.4595185996 | 30.4630926331 | 6 |
| 3 | 30.4630926331 | 30.4630922138 | 30.4630924235 | 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 √928 = 30.4630924235 to every decimal shown.
√928 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √928 the pattern is [30; 2, 6, 3, 1, 1, 1, 8, 15, 8, 1, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √928 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 4.6 × 10⁻¹ |
| 61/2 | 30.5000000000 | 3.7 × 10⁻² |
| 396/13 | 30.4615384615 | 1.6 × 10⁻³ |
| 1,249/41 | 30.4634146341 | 3.2 × 10⁻⁴ |
| 1,645/54 | 30.4629629630 | 1.3 × 10⁻⁴ |
| 2,894/95 | 30.4631578947 | 6.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 928y² = 1. Its smallest solution in positive whole numbers is x = 768,555,217, y = 25,229,061.
√928 in geometry and everyday measurements
- 928 square feet is 86.2 m². Laid out as a square — a small house footprint or a lot — it is about 30.46 ft (30 ft 6 in) on a side.
- 928 = 12² + 28², so by the Pythagorean theorem √928 is the diagonal of a 12 × 28 rectangle — and the distance between the points (0, 0) and (12, 28) on a grid.
- Since √928 = 4√58, a length of √928 is exactly 4 copies of the length √58 laid end to end.
Square roots near √928 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √925 | 5√37 | 30.4138 | No |
| √926 | √926 | 30.4302 | No |
| √927 | 3√103 | 30.4467 | No |
| √928 | 4√58 | 30.4631 | No |
| √929 | √929 | 30.4795 | No |
| √930 | √930 | 30.4959 | No |
| √931 | 7√19 | 30.5123 | No |
- The cube root of 928 is about 9.753998.
- Because 928 = 4 × 232, the root is twice √232: 2 × 15.231546 ≈ 30.463092.
Frequently asked questions
What is the square root of 928?
The square root of 928 is 4√58 in simplest radical form, which is about 30.4630924235. The negative root, −30.463092, also squares to 928.
Is the square root of 928 rational or irrational?
Irrational. 928 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √928 be simplified?
Yes. The largest perfect square dividing 928 is 16, so √928 = √16 × √58 = 4√58.
What is √928 rounded to two decimal places?
√928 ≈ 30.46 to two decimal places (30.5 to one, 30.463 to three). Check: 30.46² = 927.8116, close to 928.