√737 at a glance
- Exact value
- √737
- Decimal (10 places)
- 27.1477439210
- Rounded
- 27.1 · 27.15 · 27.148
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.147744
- Prime factorization
- 11 × 67
- Cube root
- 9.032802
How to simplify √737
The prime factorization of 737 is 11 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √737 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 737, 11 and 67 appear an odd number of times, so √737 is irrational and 27.1477439210 is a rounded value.
Where √737 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √737 lies between 27 and 28. 737 is 8 above 729 and 47 below 784, so the root is closer to 27.
- Straight line between 729 and 784: 27.1455 (0.01% low)
- Tangent from 27, i.e. 27 + 8 ÷ 54: 27.1481 (0% high)
- Tangent from 28, i.e. 28 − 47 ÷ 56: 27.1607 (0.05% high)
For √737 the tangent at 27 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 737 is just 8 above 729.
Finding √737 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 737: following the tangent line down to zero simplifies to averaging x with 737 ÷ x.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 737 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 27.2962962963 | 27.1481481481 | 3 |
| 2 | 27.1481481481 | 27.1473396999 | 27.1477439240 | 8 |
| 3 | 27.1477439240 | 27.1477439180 | 27.1477439210 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √737 = 27.1477439210 to every decimal shown.
√737 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √737 the pattern is [27; 6, 1, 3, 3, 7, 2, 4, 2, 7, 3, 3, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √737 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 | 1.5 × 10⁻¹ |
| 163/6 | 27.1666666667 | 1.9 × 10⁻² |
| 190/7 | 27.1428571429 | 4.9 × 10⁻³ |
| 733/27 | 27.1481481481 | 4.0 × 10⁻⁴ |
| 2,389/88 | 27.1477272727 | 1.7 × 10⁻⁵ |
| 17,456/643 | 27.1477449456 | 1.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 737y² = 1. Its smallest solution in positive whole numbers is x = 252,975,383, y = 9,318,468.
√737 in geometry and everyday measurements
- 737 square feet is 68.5 m². Laid out as a square — a small house footprint or a lot — it is about 27.15 ft (27 ft 2 in) on a side.
- 737 is not a sum of two whole-number squares — the prime factor 11 and 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √737 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 27 box, because 2² + 2² + 27² = 737.
Square roots near √737 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √734 | √734 | 27.0924 | No |
| √735 | 7√15 | 27.1109 | No |
| √736 | 4√46 | 27.1293 | No |
| √737 | √737 | 27.1477 | No |
| √738 | 3√82 | 27.1662 | No |
| √739 | √739 | 27.1846 | No |
| √740 | 2√185 | 27.2029 | No |
- The cube root of 737 is about 9.032802.
- Squaring undoes the root: (√737)² = 737, while 737² = 543,169 — the number whose square root is 737.
Frequently asked questions
What is the square root of 737?
The square root of 737 is √737, about 27.1477439210. The negative root, −27.147744, also squares to 737.
Is the square root of 737 rational or irrational?
Irrational. 737 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 √737 be simplified?
No. 737 = 11 × 67 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √737 rounded to two decimal places?
√737 ≈ 27.15 to two decimal places (27.1 to one, 27.148 to three). Check: 27.15² = 737.1225, close to 737.