√734 at a glance
- Exact value
- √734
- Decimal (10 places)
- 27.0924343683
- Rounded
- 27.1 · 27.09 · 27.092
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.092434
- Prime factorization
- 2 × 367
- Cube root
- 9.020529
How to simplify √734
The prime factorization of 734 is 2 × 367. Every prime appears only once, so there is no pair to bring outside the radical — √734 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 734, 2 and 367 appear an odd number of times, so √734 is irrational and 27.0924343683 is a rounded value.
Where √734 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √734 lies between 27 and 28. 734 is 5 above 729 and 50 below 784, so the root is closer to 27.
- Straight line between 729 and 784: 27.0909 (0.01% low)
- Tangent from 27, i.e. 27 + 5 ÷ 54: 27.0926 (0% high)
- Tangent from 28, i.e. 28 − 50 ÷ 56: 27.1071 (0.05% high)
For √734 the tangent at 27 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 734 is just 5 above 729.
Finding √734 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, 27 (27² = 729):
| Step | Guess x | 734 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 27.1851851852 | 27.0925925926 | 3 |
| 2 | 27.0925925926 | 27.0922761449 | 27.0924343688 | 9 |
| 3 | 27.0924343688 | 27.0924343678 | 27.0924343683 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √734 = 27.0924343683 to every decimal shown.
√734 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √734 the pattern is [27; 10, 1, 4, 1, 1, 26, 1, 1, 4, 1, 10, 54] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √734 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 | 9.2 × 10⁻² |
| 271/10 | 27.1000000000 | 7.6 × 10⁻³ |
| 298/11 | 27.0909090909 | 1.5 × 10⁻³ |
| 1,463/54 | 27.0925925926 | 1.6 × 10⁻⁴ |
| 1,761/65 | 27.0923076923 | 1.3 × 10⁻⁴ |
| 3,224/119 | 27.0924369748 | 2.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 734y² = 1. Its smallest solution in positive whole numbers is x = 10,394,175, y = 383,656.
√734 in geometry and everyday measurements
- 734 square feet is 68.2 m². Laid out as a square — a small house footprint or a lot — it is about 27.09 ft (27 ft 1 in) on a side.
- 734 is not a sum of two whole-number squares — the prime factor 367 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √734 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 27 box, because 1² + 2² + 27² = 734.
Square roots near √734 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √731 | √731 | 27.0370 | No |
| √732 | 2√183 | 27.0555 | No |
| √733 | √733 | 27.0740 | No |
| √734 | √734 | 27.0924 | No |
| √735 | 7√15 | 27.1109 | No |
| √736 | 4√46 | 27.1293 | No |
| √737 | √737 | 27.1477 | No |
- The cube root of 734 is about 9.020529.
- Squaring undoes the root: (√734)² = 734, while 734² = 538,756 — the number whose square root is 734.
Frequently asked questions
What is the square root of 734?
The square root of 734 is √734, about 27.0924343683. The negative root, −27.092434, also squares to 734.
Is the square root of 734 rational or irrational?
Irrational. 734 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 √734 be simplified?
No. 734 = 2 × 367 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √734 rounded to two decimal places?
√734 ≈ 27.09 to two decimal places (27.1 to one, 27.092 to three). Check: 27.09² = 733.8681, close to 734.