√715 at a glance
- Exact value
- √715
- Decimal (10 places)
- 26.7394839142
- Rounded
- 26.7 · 26.74 · 26.739
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.739484
- Prime factorization
- 5 × 11 × 13
- Cube root
- 8.942014
How to simplify √715
The prime factorization of 715 is 5 × 11 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √715 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 715, 5, 11 and 13 appear an odd number of times, so √715 is irrational and 26.7394839142 is a rounded value.
Where √715 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √715 lies between 26 and 27. 715 is 39 above 676 and 14 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.7358 (0.01% low)
- Tangent from 26, i.e. 26 + 39 ÷ 52: 26.7500 (0.04% high)
- Tangent from 27, i.e. 27 − 14 ÷ 54: 26.7407 (0% high)
For √715 the tangent at 27 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 715 is just 14 below 729.
Finding √715 with the Babylonian method
If a guess is too big, 715 ÷ 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√715) in one step.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 715 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.4814814815 | 26.7407407407 | 2 |
| 2 | 26.7407407407 | 26.7382271468 | 26.7394839438 | 7 |
| 3 | 26.7394839438 | 26.7394838847 | 26.7394839142 | 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 √715 = 26.7394839142 to every decimal shown.
√715 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √715 the pattern is [26; 1, 2, 1, 5, 5, 5, 1, 2, 1, 52] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √715 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 | 7.4 × 10⁻¹ |
| 27/1 | 27.0000000000 | 2.6 × 10⁻¹ |
| 80/3 | 26.6666666667 | 7.3 × 10⁻² |
| 107/4 | 26.7500000000 | 1.1 × 10⁻² |
| 615/23 | 26.7391304348 | 3.5 × 10⁻⁴ |
| 3,182/119 | 26.7394957983 | 1.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 715y² = 1. Its smallest solution in positive whole numbers is x = 75,646, y = 2,829.
√715 in geometry and everyday measurements
- 715 square feet is 66.4 m². Laid out as a square — a small house footprint or a lot — it is about 26.74 ft (26 ft 9 in) on a side.
- 715 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √715 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 9 × 25 box, because 3² + 9² + 25² = 715.
Square roots near √715 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √712 | 2√178 | 26.6833 | No |
| √713 | √713 | 26.7021 | No |
| √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 |
- The cube root of 715 is about 8.942014.
- Squaring undoes the root: (√715)² = 715, while 715² = 511,225 — the number whose square root is 715.
Frequently asked questions
What is the square root of 715?
The square root of 715 is √715, about 26.7394839142. The negative root, −26.739484, also squares to 715.
Is the square root of 715 rational or irrational?
Irrational. 715 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 √715 be simplified?
No. 715 = 5 × 11 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √715 rounded to two decimal places?
√715 ≈ 26.74 to two decimal places (26.7 to one, 26.739 to three). Check: 26.74² = 715.0276, close to 715.