√414 at a glance
- Exact value
- 3√46
- Decimal (10 places)
- 20.3469899494
- Rounded
- 20.3 · 20.35 · 20.347
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.346990
- Prime factorization
- 2 × 3² × 23
- Cube root
- 7.453040
How to simplify √414
Look for the largest perfect square that divides 414. Here it is 9 (3²), because 414 = 9 × 46 and 46 has no square factor left:
The prime factorization tells the same story: 414 = 2 × 3² × 23. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 23 stays inside.
Check: (3√46)² = 3² × 46 = 9 × 46 = 414. As a decimal, 3√46 = 3 × 6.7823299831 ≈ 20.3469899494.
Where √414 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √414 lies between 20 and 21. 414 is 14 above 400 and 27 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.3415 (0.03% low)
- Tangent from 20, i.e. 20 + 14 ÷ 40: 20.3500 (0.01% high)
- Tangent from 21, i.e. 21 − 27 ÷ 42: 20.3571 (0.05% high)
For √414 the tangent at 20 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 414 is just 14 above 400.
Finding √414 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 | 414 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.7000000000 | 20.3500000000 | 2 |
| 2 | 20.3500000000 | 20.3439803440 | 20.3469901720 | 6 |
| 3 | 20.3469901720 | 20.3469897268 | 20.3469899494 | 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 √414 = 20.3469899494 to every decimal shown.
√414 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √414 the pattern is [20; 2, 1, 7, 2, 7, 1, 2, 40] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √414 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 | 3.5 × 10⁻¹ |
| 41/2 | 20.5000000000 | 1.5 × 10⁻¹ |
| 61/3 | 20.3333333333 | 1.4 × 10⁻² |
| 468/23 | 20.3478260870 | 8.4 × 10⁻⁴ |
| 997/49 | 20.3469387755 | 5.1 × 10⁻⁵ |
| 7,447/366 | 20.3469945355 | 4.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 414y² = 1. Its smallest solution in positive whole numbers is x = 24,335, y = 1,196.
√414 in geometry and everyday measurements
- A square garage floor of 414 square feet measures about 20.35 ft (20 ft 4 in) per side, and its corner-to-corner diagonal is √828 ≈ 28.8 ft.
- 414 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √414 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 19 box, because 2² + 7² + 19² = 414.
- Since √414 = 3√46, a length of √414 is exactly 3 copies of the length √46 laid end to end.
Square roots near √414 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √411 | √411 | 20.2731 | No |
| √412 | 2√103 | 20.2978 | No |
| √413 | √413 | 20.3224 | No |
| √414 | 3√46 | 20.3470 | No |
| √415 | √415 | 20.3715 | No |
| √416 | 4√26 | 20.3961 | No |
| √417 | √417 | 20.4206 | No |
- The cube root of 414 is about 7.453040.
- Squaring undoes the root: (√414)² = 414, while 414² = 171,396 — the number whose square root is 414.
Frequently asked questions
What is the square root of 414?
The square root of 414 is 3√46 in simplest radical form, which is about 20.3469899494. The negative root, −20.346990, also squares to 414.
Is the square root of 414 rational or irrational?
Irrational. 414 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 √414 be simplified?
Yes. The largest perfect square dividing 414 is 9, so √414 = √9 × √46 = 3√46.
What is √414 rounded to two decimal places?
√414 ≈ 20.35 to two decimal places (20.3 to one, 20.347 to three). Check: 20.35² = 414.1225, close to 414.