√430 at a glance
- Exact value
- √430
- Decimal (10 places)
- 20.7364413533
- Rounded
- 20.7 · 20.74 · 20.736
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.736441
- Prime factorization
- 2 × 5 × 43
- Cube root
- 7.547842
How to simplify √430
The prime factorization of 430 is 2 × 5 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √430 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 430, 2, 5 and 43 appear an odd number of times, so √430 is irrational and 20.7364413533 is a rounded value.
Where √430 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √430 lies between 20 and 21. 430 is 30 above 400 and 11 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.7317 (0.02% low)
- Tangent from 20, i.e. 20 + 30 ÷ 40: 20.7500 (0.07% high)
- Tangent from 21, i.e. 21 − 11 ÷ 42: 20.7381 (0.01% high)
For √430 the tangent at 21 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 430 is just 11 below 441.
Finding √430 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 | 430 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.4761904762 | 20.7380952381 | 2 |
| 2 | 20.7380952381 | 20.7347876005 | 20.7364414193 | 7 |
| 3 | 20.7364414193 | 20.7364412874 | 20.7364413533 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √430 = 20.7364413533 to every decimal shown.
√430 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √430 the pattern is [20; 1, 2, 1, 3, 1, 6, 8, 6, 1, 3, 1, 2, …] 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 √430 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 | 7.4 × 10⁻¹ |
| 21/1 | 21.0000000000 | 2.6 × 10⁻¹ |
| 62/3 | 20.6666666667 | 7.0 × 10⁻² |
| 83/4 | 20.7500000000 | 1.4 × 10⁻² |
| 311/15 | 20.7333333333 | 3.1 × 10⁻³ |
| 394/19 | 20.7368421053 | 4.0 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 430y² = 1. Its smallest solution in positive whole numbers is x = 2,862,251, y = 138,030.
√430 in geometry and everyday measurements
- A square garage floor of 430 square feet measures about 20.74 ft (20 ft 9 in) per side, and its corner-to-corner diagonal is √860 ≈ 29.3 ft.
- 430 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √430 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 14 × 15 box, because 3² + 14² + 15² = 430.
Square roots near √430 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √427 | √427 | 20.6640 | No |
| √428 | 2√107 | 20.6882 | No |
| √429 | √429 | 20.7123 | No |
| √430 | √430 | 20.7364 | No |
| √431 | √431 | 20.7605 | No |
| √432 | 12√3 | 20.7846 | No |
| √433 | √433 | 20.8087 | No |
- The cube root of 430 is about 7.547842.
- Squaring undoes the root: (√430)² = 430, while 430² = 184,900 — the number whose square root is 430.
Frequently asked questions
What is the square root of 430?
The square root of 430 is √430, about 20.7364413533. The negative root, −20.736441, also squares to 430.
Is the square root of 430 rational or irrational?
Irrational. 430 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 √430 be simplified?
No. 430 = 2 × 5 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √430 rounded to two decimal places?
√430 ≈ 20.74 to two decimal places (20.7 to one, 20.736 to three). Check: 20.74² = 430.1476, close to 430.