√759 at a glance
- Exact value
- √759
- Decimal (10 places)
- 27.5499546279
- Rounded
- 27.5 · 27.55 · 27.550
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.549955
- Prime factorization
- 3 × 11 × 23
- Cube root
- 9.121801
How to simplify √759
The prime factorization of 759 is 3 × 11 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √759 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 759, 3, 11 and 23 appear an odd number of times, so √759 is irrational and 27.5499546279 is a rounded value.
Where √759 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √759 lies between 27 and 28. 759 is 30 above 729 and 25 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.5455 (0.02% low)
- Tangent from 27, i.e. 27 + 30 ÷ 54: 27.5556 (0.02% high)
- Tangent from 28, i.e. 28 − 25 ÷ 56: 27.5536 (0.01% high)
For √759 the tangent at 28 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 759 is just 25 below 784.
Finding √759 with the Babylonian method
If a guess is too big, 759 ÷ 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√759) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 759 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.1071428571 | 27.5535714286 | 2 |
| 2 | 27.5535714286 | 27.5463383020 | 27.5499548653 | 6 |
| 3 | 27.5499548653 | 27.5499543905 | 27.5499546279 | 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 √759 = 27.5499546279 to every decimal shown.
√759 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √759 the pattern is [27; 1, 1, 4, 1, 1, 54] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √759 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 |
|---|---|---|
| 27/1 | 27.0000000000 | 5.5 × 10⁻¹ |
| 28/1 | 28.0000000000 | 4.5 × 10⁻¹ |
| 55/2 | 27.5000000000 | 5.0 × 10⁻² |
| 248/9 | 27.5555555556 | 5.6 × 10⁻³ |
| 303/11 | 27.5454545455 | 4.5 × 10⁻³ |
| 551/20 | 27.5500000000 | 4.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 759y² = 1. Its smallest solution in positive whole numbers is x = 551, y = 20.
√759 in geometry and everyday measurements
- 759 square feet is 70.5 m². Laid out as a square — a small house footprint or a lot — it is about 27.55 ft (27 ft 7 in) on a side.
- 759 is not a sum of two whole-number squares — the prime factor 3, 11 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √759 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √759 as its space diagonal.
Square roots near √759 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √756 | 6√21 | 27.4955 | No |
| √757 | √757 | 27.5136 | No |
| √758 | √758 | 27.5318 | No |
| √759 | √759 | 27.5500 | No |
| √760 | 2√190 | 27.5681 | No |
| √761 | √761 | 27.5862 | No |
| √762 | √762 | 27.6043 | No |
- The cube root of 759 is about 9.121801.
- Squaring undoes the root: (√759)² = 759, while 759² = 576,081 — the number whose square root is 759.
Frequently asked questions
What is the square root of 759?
The square root of 759 is √759, about 27.5499546279. The negative root, −27.549955, also squares to 759.
Is the square root of 759 rational or irrational?
Irrational. 759 is not a perfect square — it falls between 729 and 784 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √759 be simplified?
No. 759 = 3 × 11 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √759 rounded to two decimal places?
√759 ≈ 27.55 to two decimal places (27.5 to one, 27.550 to three). Check: 27.55² = 759.0025, close to 759.