√470 at a glance
- Exact value
- √470
- Decimal (10 places)
- 21.6794833887
- Rounded
- 21.7 · 21.68 · 21.679
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.679483
- Prime factorization
- 2 × 5 × 47
- Cube root
- 7.774980
How to simplify √470
The prime factorization of 470 is 2 × 5 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √470 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 470, 2, 5 and 47 appear an odd number of times, so √470 is irrational and 21.6794833887 is a rounded value.
Where √470 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √470 lies between 21 and 22. 470 is 29 above 441 and 14 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.6744 (0.02% low)
- Tangent from 21, i.e. 21 + 29 ÷ 42: 21.6905 (0.05% high)
- Tangent from 22, i.e. 22 − 14 ÷ 44: 21.6818 (0.01% high)
For √470 the tangent at 22 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 470 is just 14 below 484.
Finding √470 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 | 470 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.3636363636 | 21.6818181818 | 2 |
| 2 | 21.6818181818 | 21.6771488470 | 21.6794835144 | 6 |
| 3 | 21.6794835144 | 21.6794832630 | 21.6794833887 | 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 √470 = 21.6794833887 to every decimal shown.
√470 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √470 the pattern is [21; 1, 2, 8, 2, 1, 42] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √470 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 | 6.8 × 10⁻¹ |
| 22/1 | 22.0000000000 | 3.2 × 10⁻¹ |
| 65/3 | 21.6666666667 | 1.3 × 10⁻² |
| 542/25 | 21.6800000000 | 5.2 × 10⁻⁴ |
| 1,149/53 | 21.6792452830 | 2.4 × 10⁻⁴ |
| 1,691/78 | 21.6794871795 | 3.8 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 470y² = 1. Its smallest solution in positive whole numbers is x = 1,691, y = 78.
√470 in geometry and everyday measurements
- A square garage floor of 470 square feet measures about 21.68 ft (21 ft 8 in) per side, and its corner-to-corner diagonal is √940 ≈ 30.7 ft.
- 470 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √470 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 21 box, because 2² + 5² + 21² = 470.
Square roots near √470 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √467 | √467 | 21.6102 | No |
| √468 | 6√13 | 21.6333 | No |
| √469 | √469 | 21.6564 | No |
| √470 | √470 | 21.6795 | No |
| √471 | √471 | 21.7025 | No |
| √472 | 2√118 | 21.7256 | No |
| √473 | √473 | 21.7486 | No |
- The cube root of 470 is about 7.774980.
- Squaring undoes the root: (√470)² = 470, while 470² = 220,900 — the number whose square root is 470.
Frequently asked questions
What is the square root of 470?
The square root of 470 is √470, about 21.6794833887. The negative root, −21.679483, also squares to 470.
Is the square root of 470 rational or irrational?
Irrational. 470 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 √470 be simplified?
No. 470 = 2 × 5 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √470 rounded to two decimal places?
√470 ≈ 21.68 to two decimal places (21.7 to one, 21.679 to three). Check: 21.68² = 470.0224, close to 470.