√758 at a glance
- Exact value
- √758
- Decimal (10 places)
- 27.5317997959
- Rounded
- 27.5 · 27.53 · 27.532
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.531800
- Prime factorization
- 2 × 379
- Cube root
- 9.117793
How to simplify √758
The prime factorization of 758 is 2 × 379. Every prime appears only once, so there is no pair to bring outside the radical — √758 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 758, 2 and 379 appear an odd number of times, so √758 is irrational and 27.5317997959 is a rounded value.
Where √758 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √758 lies between 27 and 28. 758 is 29 above 729 and 26 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.5273 (0.02% low)
- Tangent from 27, i.e. 27 + 29 ÷ 54: 27.5370 (0.02% high)
- Tangent from 28, i.e. 28 − 26 ÷ 56: 27.5357 (0.01% high)
For √758 the tangent at 28 wins, missing by only 0.0039. Tangent estimates shine when the number sits close to a perfect square — here 758 is just 26 below 784.
Finding √758 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 758 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.0714285714 | 27.5357142857 | 2 |
| 2 | 27.5357142857 | 27.5278858625 | 27.5318000741 | 6 |
| 3 | 27.5318000741 | 27.5317995176 | 27.5317997959 | 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 √758 = 27.5317997959 to every decimal shown.
√758 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √758 the pattern is [27; 1, 1, 7, 2, 1, 3, 3, 1, 26, 1, 3, 3, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √758 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 | 5.3 × 10⁻¹ |
| 28/1 | 28.0000000000 | 4.7 × 10⁻¹ |
| 55/2 | 27.5000000000 | 3.2 × 10⁻² |
| 413/15 | 27.5333333333 | 1.5 × 10⁻³ |
| 881/32 | 27.5312500000 | 5.5 × 10⁻⁴ |
| 1,294/47 | 27.5319148936 | 1.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 758y² = 1. Its smallest solution in positive whole numbers is x = 413,959,717, y = 15,035,694.
√758 in geometry and everyday measurements
- 758 square feet is 70.4 m². Laid out as a square — a small house footprint or a lot — it is about 27.53 ft (27 ft 6 in) on a side.
- 758 is not a sum of two whole-number squares — the prime factor 379 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √758 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 26 box, because 1² + 9² + 26² = 758.
Square roots near √758 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √755 | √755 | 27.4773 | No |
| √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 |
- The cube root of 758 is about 9.117793.
- Squaring undoes the root: (√758)² = 758, while 758² = 574,564 — the number whose square root is 758.
Frequently asked questions
What is the square root of 758?
The square root of 758 is √758, about 27.5317997959. The negative root, −27.531800, also squares to 758.
Is the square root of 758 rational or irrational?
Irrational. 758 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 √758 be simplified?
No. 758 = 2 × 379 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √758 rounded to two decimal places?
√758 ≈ 27.53 to two decimal places (27.5 to one, 27.532 to three). Check: 27.53² = 757.9009, close to 758.