√438 at a glance
- Exact value
- √438
- Decimal (10 places)
- 20.9284495365
- Rounded
- 20.9 · 20.93 · 20.928
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.928450
- Prime factorization
- 2 × 3 × 73
- Cube root
- 7.594363
How to simplify √438
The prime factorization of 438 is 2 × 3 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √438 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 438, 2, 3 and 73 appear an odd number of times, so √438 is irrational and 20.9284495365 is a rounded value.
Where √438 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √438 lies between 20 and 21. 438 is 38 above 400 and 3 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.9268 (0.01% low)
- Tangent from 20, i.e. 20 + 38 ÷ 40: 20.9500 (0.1% high)
- Tangent from 21, i.e. 21 − 3 ÷ 42: 20.9286 (0% high)
For √438 the tangent at 21 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 438 is just 3 below 441.
Finding √438 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, 21 (21² = 441):
| Step | Guess x | 438 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.8571428571 | 20.9285714286 | 3 |
| 2 | 20.9285714286 | 20.9283276451 | 20.9284495368 | 9 |
| 3 | 20.9284495368 | 20.9284495361 | 20.9284495365 | 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 √438 = 20.9284495365 to every decimal shown.
√438 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √438 the pattern is [20; 1, 12, 1, 40] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √438 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 9.3 × 10⁻¹ |
| 21/1 | 21.0000000000 | 7.2 × 10⁻² |
| 272/13 | 20.9230769231 | 5.4 × 10⁻³ |
| 293/14 | 20.9285714286 | 1.2 × 10⁻⁴ |
| 11,992/573 | 20.9284467714 | 2.8 × 10⁻⁶ |
| 12,285/587 | 20.9284497445 | 2.1 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 438y² = 1. Its smallest solution in positive whole numbers is x = 293, y = 14.
√438 in geometry and everyday measurements
- A square garage floor of 438 square feet measures about 20.93 ft (20 ft 11 in) per side, and its corner-to-corner diagonal is √876 ≈ 29.6 ft.
- 438 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √438 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 7 × 10 × 17 box, because 7² + 10² + 17² = 438.
Square roots near √438 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √435 | √435 | 20.8567 | No |
| √436 | 2√109 | 20.8806 | No |
| √437 | √437 | 20.9045 | No |
| √438 | √438 | 20.9284 | No |
| √439 | √439 | 20.9523 | No |
| √440 | 2√110 | 20.9762 | No |
| √441 | 21 | 21.0000 | Yes |
- The cube root of 438 is about 7.594363.
- Squaring undoes the root: (√438)² = 438, while 438² = 191,844 — the number whose square root is 438.
Frequently asked questions
What is the square root of 438?
The square root of 438 is √438, about 20.9284495365. The negative root, −20.928450, also squares to 438.
Is the square root of 438 rational or irrational?
Irrational. 438 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √438 be simplified?
No. 438 = 2 × 3 × 73 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √438 rounded to two decimal places?
√438 ≈ 20.93 to two decimal places (20.9 to one, 20.928 to three). Check: 20.93² = 438.0649, close to 438.