√917 at a glance
- Exact value
- √917
- Decimal (10 places)
- 30.2820078595
- Rounded
- 30.3 · 30.28 · 30.282
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.282008
- Prime factorization
- 7 × 131
- Cube root
- 9.715305
How to simplify √917
The prime factorization of 917 is 7 × 131. Every prime appears only once, so there is no pair to bring outside the radical — √917 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 917, 7 and 131 appear an odd number of times, so √917 is irrational and 30.2820078595 is a rounded value.
Where √917 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √917 lies between 30 and 31. 917 is 17 above 900 and 44 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.2787 (0.01% low)
- Tangent from 30, i.e. 30 + 17 ÷ 60: 30.2833 (0% high)
- Tangent from 31, i.e. 31 − 44 ÷ 62: 30.2903 (0.03% high)
For √917 the tangent at 30 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 917 is just 17 above 900.
Finding √917 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 917: following the tangent line down to zero simplifies to averaging x with 917 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 917 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.5666666667 | 30.2833333333 | 2 |
| 2 | 30.2833333333 | 30.2806824436 | 30.2820078885 | 7 |
| 3 | 30.2820078885 | 30.2820078304 | 30.2820078595 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √917 = 30.2820078595 to every decimal shown.
√917 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √917 the pattern is [30; 3, 1, 1, 4, 1, 14, 3, 8, 3, 14, 1, 4, …] 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 √917 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 | 2.8 × 10⁻¹ |
| 91/3 | 30.3333333333 | 5.1 × 10⁻² |
| 121/4 | 30.2500000000 | 3.2 × 10⁻² |
| 212/7 | 30.2857142857 | 3.7 × 10⁻³ |
| 969/32 | 30.2812500000 | 7.6 × 10⁻⁴ |
| 1,181/39 | 30.2820512821 | 4.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 917y² = 1. Its smallest solution in positive whole numbers is x = 823,604,599, y = 27,197,820.
√917 in geometry and everyday measurements
- 917 square feet is 85.2 m². Laid out as a square — a small house footprint or a lot — it is about 30.28 ft (30 ft 3 in) on a side.
- 917 is not a sum of two whole-number squares — the prime factor 7 and 131 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √917 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 30 box, because 1² + 4² + 30² = 917.
Square roots near √917 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √914 | √914 | 30.2324 | No |
| √915 | √915 | 30.2490 | No |
| √916 | 2√229 | 30.2655 | No |
| √917 | √917 | 30.2820 | No |
| √918 | 3√102 | 30.2985 | No |
| √919 | √919 | 30.3150 | No |
| √920 | 2√230 | 30.3315 | No |
- The cube root of 917 is about 9.715305.
- Squaring undoes the root: (√917)² = 917, while 917² = 840,889 — the number whose square root is 917.
Frequently asked questions
What is the square root of 917?
The square root of 917 is √917, about 30.2820078595. The negative root, −30.282008, also squares to 917.
Is the square root of 917 rational or irrational?
Irrational. 917 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 √917 be simplified?
No. 917 = 7 × 131 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √917 rounded to two decimal places?
√917 ≈ 30.28 to two decimal places (30.3 to one, 30.282 to three). Check: 30.28² = 916.8784, close to 917.