√717 at a glance
- Exact value
- √717
- Decimal (10 places)
- 26.7768556780
- Rounded
- 26.8 · 26.78 · 26.777
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.776856
- Prime factorization
- 3 × 239
- Cube root
- 8.950344
How to simplify √717
The prime factorization of 717 is 3 × 239. Every prime appears only once, so there is no pair to bring outside the radical — √717 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 717, 3 and 239 appear an odd number of times, so √717 is irrational and 26.7768556780 is a rounded value.
Where √717 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √717 lies between 26 and 27. 717 is 41 above 676 and 12 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.7736 (0.01% low)
- Tangent from 26, i.e. 26 + 41 ÷ 52: 26.7885 (0.04% high)
- Tangent from 27, i.e. 27 − 12 ÷ 54: 26.7778 (0% high)
For √717 the tangent at 27 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 717 is just 12 below 729.
Finding √717 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 717: following the tangent line down to zero simplifies to averaging x with 717 ÷ x.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 717 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.5555555556 | 26.7777777778 | 3 |
| 2 | 26.7777777778 | 26.7759336100 | 26.7768556939 | 7 |
| 3 | 26.7768556939 | 26.7768556621 | 26.7768556780 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √717 = 26.7768556780 to every decimal shown.
√717 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √717 the pattern is [26; 1, 3, 2, 12, 1, 16, 1, 12, 2, 3, 1, 52] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √717 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 |
|---|---|---|
| 26/1 | 26.0000000000 | 7.8 × 10⁻¹ |
| 27/1 | 27.0000000000 | 2.2 × 10⁻¹ |
| 107/4 | 26.7500000000 | 2.7 × 10⁻² |
| 241/9 | 26.7777777778 | 9.2 × 10⁻⁴ |
| 2,999/112 | 26.7767857143 | 7.0 × 10⁻⁵ |
| 3,240/121 | 26.7768595041 | 3.8 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 717y² = 1. Its smallest solution in positive whole numbers is x = 6,998,399, y = 261,360.
√717 in geometry and everyday measurements
- 717 square feet is 66.6 m². Laid out as a square — a small house footprint or a lot — it is about 26.78 ft (26 ft 9 in) on a side.
- 717 is not a sum of two whole-number squares — the prime factor 3 and 239 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √717 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 5 × 26 box, because 4² + 5² + 26² = 717.
Square roots near √717 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √714 | √714 | 26.7208 | No |
| √715 | √715 | 26.7395 | No |
| √716 | 2√179 | 26.7582 | No |
| √717 | √717 | 26.7769 | No |
| √718 | √718 | 26.7955 | No |
| √719 | √719 | 26.8142 | No |
| √720 | 12√5 | 26.8328 | No |
- The cube root of 717 is about 8.950344.
- Squaring undoes the root: (√717)² = 717, while 717² = 514,089 — the number whose square root is 717.
Frequently asked questions
What is the square root of 717?
The square root of 717 is √717, about 26.7768556780. The negative root, −26.776856, also squares to 717.
Is the square root of 717 rational or irrational?
Irrational. 717 is not a perfect square — it falls between 676 and 729 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √717 be simplified?
No. 717 = 3 × 239 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √717 rounded to two decimal places?
√717 ≈ 26.78 to two decimal places (26.8 to one, 26.777 to three). Check: 26.78² = 717.1684, close to 717.