√434 at a glance
- Exact value
- √434
- Decimal (10 places)
- 20.8326666560
- Rounded
- 20.8 · 20.83 · 20.833
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.832667
- Prime factorization
- 2 × 7 × 31
- Cube root
- 7.571174
How to simplify √434
The prime factorization of 434 is 2 × 7 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √434 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 434, 2, 7 and 31 appear an odd number of times, so √434 is irrational and 20.8326666560 is a rounded value.
Where √434 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √434 lies between 20 and 21. 434 is 34 above 400 and 7 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.8293 (0.02% low)
- Tangent from 20, i.e. 20 + 34 ÷ 40: 20.8500 (0.08% high)
- Tangent from 21, i.e. 21 − 7 ÷ 42: 20.8333 (0% high)
For √434 the tangent at 21 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 434 is just 7 below 441.
Finding √434 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 | 434 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.6666666667 | 20.8333333333 | 3 |
| 2 | 20.8333333333 | 20.8320000000 | 20.8326666667 | 7 |
| 3 | 20.8326666667 | 20.8326666453 | 20.8326666560 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √434 = 20.8326666560 to every decimal shown.
√434 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √434 the pattern is [20; 1, 4, 1, 40] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √434 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 | 8.3 × 10⁻¹ |
| 21/1 | 21.0000000000 | 1.7 × 10⁻¹ |
| 104/5 | 20.8000000000 | 3.3 × 10⁻² |
| 125/6 | 20.8333333333 | 6.7 × 10⁻⁴ |
| 5,104/245 | 20.8326530612 | 1.4 × 10⁻⁵ |
| 5,229/251 | 20.8326693227 | 2.7 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 434y² = 1. Its smallest solution in positive whole numbers is x = 125, y = 6.
√434 in geometry and everyday measurements
- A square garage floor of 434 square feet measures about 20.83 ft (20 ft 10 in) per side, and its corner-to-corner diagonal is √868 ≈ 29.5 ft.
- 434 is not a sum of two whole-number squares — the prime factor 7 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √434 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 12 × 17 box, because 1² + 12² + 17² = 434.
Square roots near √434 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √431 | √431 | 20.7605 | No |
| √432 | 12√3 | 20.7846 | No |
| √433 | √433 | 20.8087 | No |
| √434 | √434 | 20.8327 | No |
| √435 | √435 | 20.8567 | No |
| √436 | 2√109 | 20.8806 | No |
| √437 | √437 | 20.9045 | No |
- The cube root of 434 is about 7.571174.
- Squaring undoes the root: (√434)² = 434, while 434² = 188,356 — the number whose square root is 434.
Frequently asked questions
What is the square root of 434?
The square root of 434 is √434, about 20.8326666560. The negative root, −20.832667, also squares to 434.
Is the square root of 434 rational or irrational?
Irrational. 434 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 √434 be simplified?
No. 434 = 2 × 7 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √434 rounded to two decimal places?
√434 ≈ 20.83 to two decimal places (20.8 to one, 20.833 to three). Check: 20.83² = 433.8889, close to 434.