√478 at a glance
- Exact value
- √478
- Decimal (10 places)
- 21.8632111091
- Rounded
- 21.9 · 21.86 · 21.863
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.863211
- Prime factorization
- 2 × 239
- Cube root
- 7.818846
How to simplify √478
The prime factorization of 478 is 2 × 239. Every prime appears only once, so there is no pair to bring outside the radical — √478 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 478, 2 and 239 appear an odd number of times, so √478 is irrational and 21.8632111091 is a rounded value.
Where √478 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √478 lies between 21 and 22. 478 is 37 above 441 and 6 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.8605 (0.01% low)
- Tangent from 21, i.e. 21 + 37 ÷ 42: 21.8810 (0.08% high)
- Tangent from 22, i.e. 22 − 6 ÷ 44: 21.8636 (0% high)
For √478 the tangent at 22 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 478 is just 6 below 484.
Finding √478 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, 22 (22² = 484):
| Step | Guess x | 478 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.7272727273 | 21.8636363636 | 3 |
| 2 | 21.8636363636 | 21.8627858628 | 21.8632111132 | 8 |
| 3 | 21.8632111132 | 21.8632111049 | 21.8632111091 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √478 = 21.8632111091 to every decimal shown.
√478 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √478 the pattern is [21; 1, 6, 3, 4, 1, 1, 5, 1, 2, 3, 1, 1, …] with the block of 36 terms after the semicolon repeating forever (only the first 12 of the 36 are shown). A pattern that never ends is one more proof that √478 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 8.6 × 10⁻¹ |
| 22/1 | 22.0000000000 | 1.4 × 10⁻¹ |
| 153/7 | 21.8571428571 | 6.1 × 10⁻³ |
| 481/22 | 21.8636363636 | 4.3 × 10⁻⁴ |
| 2,077/95 | 21.8631578947 | 5.3 × 10⁻⁵ |
| 2,558/117 | 21.8632478632 | 3.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 478y² = 1. Its smallest solution in positive whole numbers is x = 1,617,319,577,991,743, y = 73,974,475,657,896 — 16 digits for x, even though 478 is small, which is what makes Pell’s equation famous.
√478 in geometry and everyday measurements
- A square garage floor of 478 square feet measures about 21.86 ft (21 ft 10 in) per side, and its corner-to-corner diagonal is √956 ≈ 30.9 ft.
- 478 is not a sum of two whole-number squares — the prime factor 239 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √478 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 21 box, because 1² + 6² + 21² = 478.
Square roots near √478 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √475 | 5√19 | 21.7945 | No |
| √476 | 2√119 | 21.8174 | No |
| √477 | 3√53 | 21.8403 | No |
| √478 | √478 | 21.8632 | No |
| √479 | √479 | 21.8861 | No |
| √480 | 4√30 | 21.9089 | No |
| √481 | √481 | 21.9317 | No |
- The cube root of 478 is about 7.818846.
- Squaring undoes the root: (√478)² = 478, while 478² = 228,484 — the number whose square root is 478.
Frequently asked questions
What is the square root of 478?
The square root of 478 is √478, about 21.8632111091. The negative root, −21.863211, also squares to 478.
Is the square root of 478 rational or irrational?
Irrational. 478 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √478 be simplified?
No. 478 = 2 × 239 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √478 rounded to two decimal places?
√478 ≈ 21.86 to two decimal places (21.9 to one, 21.863 to three). Check: 21.86² = 477.8596, close to 478.