√418 at a glance
- Exact value
- √418
- Decimal (10 places)
- 20.4450483003
- Rounded
- 20.4 · 20.45 · 20.445
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.445048
- Prime factorization
- 2 × 11 × 19
- Cube root
- 7.476966
How to simplify √418
The prime factorization of 418 is 2 × 11 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √418 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 418, 2, 11 and 19 appear an odd number of times, so √418 is irrational and 20.4450483003 is a rounded value.
Where √418 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √418 lies between 20 and 21. 418 is 18 above 400 and 23 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.4390 (0.03% low)
- Tangent from 20, i.e. 20 + 18 ÷ 40: 20.4500 (0.02% high)
- Tangent from 21, i.e. 21 − 23 ÷ 42: 20.4524 (0.04% high)
For √418 the tangent at 20 wins, missing by only 0.005. Tangent estimates shine when the number sits close to a perfect square — here 418 is just 18 above 400.
Finding √418 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, 20 (20² = 400):
| Step | Guess x | 418 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.9000000000 | 20.4500000000 | 2 |
| 2 | 20.4500000000 | 20.4400977995 | 20.4450488998 | 6 |
| 3 | 20.4450488998 | 20.4450477008 | 20.4450483003 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √418 = 20.4450483003 to every decimal shown.
√418 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √418 the pattern is [20; 2, 4, 20, 4, 2, 40] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √418 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 | 4.5 × 10⁻¹ |
| 41/2 | 20.5000000000 | 5.5 × 10⁻² |
| 184/9 | 20.4444444444 | 6.0 × 10⁻⁴ |
| 3,721/182 | 20.4450549451 | 6.6 × 10⁻⁶ |
| 15,068/737 | 20.4450474898 | 8.1 × 10⁻⁷ |
| 33,857/1,656 | 20.4450483092 | 8.9 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 418y² = 1. Its smallest solution in positive whole numbers is x = 33,857, y = 1,656.
√418 in geometry and everyday measurements
- A square garage floor of 418 square feet measures about 20.45 ft (20 ft 5 in) per side, and its corner-to-corner diagonal is √836 ≈ 28.9 ft.
- 418 is not a sum of two whole-number squares — the prime factor 11 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √418 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 20 box, because 3² + 3² + 20² = 418.
Square roots near √418 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √415 | √415 | 20.3715 | No |
| √416 | 4√26 | 20.3961 | No |
| √417 | √417 | 20.4206 | No |
| √418 | √418 | 20.4450 | No |
| √419 | √419 | 20.4695 | No |
| √420 | 2√105 | 20.4939 | No |
| √421 | √421 | 20.5183 | No |
- The cube root of 418 is about 7.476966.
- Squaring undoes the root: (√418)² = 418, while 418² = 174,724 — the number whose square root is 418.
Frequently asked questions
What is the square root of 418?
The square root of 418 is √418, about 20.4450483003. The negative root, −20.445048, also squares to 418.
Is the square root of 418 rational or irrational?
Irrational. 418 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 √418 be simplified?
No. 418 = 2 × 11 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √418 rounded to two decimal places?
√418 ≈ 20.45 to two decimal places (20.4 to one, 20.445 to three). Check: 20.45² = 418.2025, close to 418.