√471 at a glance
- Exact value
- √471
- Decimal (10 places)
- 21.7025344142
- Rounded
- 21.7 · 21.70 · 21.703
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.702534
- Prime factorization
- 3 × 157
- Cube root
- 7.780490
How to simplify √471
The prime factorization of 471 is 3 × 157. Every prime appears only once, so there is no pair to bring outside the radical — √471 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 471, 3 and 157 appear an odd number of times, so √471 is irrational and 21.7025344142 is a rounded value.
Where √471 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √471 lies between 21 and 22. 471 is 30 above 441 and 13 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.6977 (0.02% low)
- Tangent from 21, i.e. 21 + 30 ÷ 42: 21.7143 (0.05% high)
- Tangent from 22, i.e. 22 − 13 ÷ 44: 21.7045 (0.01% high)
For √471 the tangent at 22 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 471 is just 13 below 484.
Finding √471 with the Babylonian method
If a guess is too big, 471 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√471) in one step.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 471 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.4090909091 | 21.7045454545 | 2 |
| 2 | 21.7045454545 | 21.7005235602 | 21.7025345074 | 7 |
| 3 | 21.7025345074 | 21.7025343210 | 21.7025344142 | 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 √471 = 21.7025344142 to every decimal shown.
√471 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √471 the pattern is [21; 1, 2, 2, 1, 3, 4, 14, 4, 3, 1, 2, 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 √471 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 | 7.0 × 10⁻¹ |
| 22/1 | 22.0000000000 | 3.0 × 10⁻¹ |
| 65/3 | 21.6666666667 | 3.6 × 10⁻² |
| 152/7 | 21.7142857143 | 1.2 × 10⁻² |
| 217/10 | 21.7000000000 | 2.5 × 10⁻³ |
| 803/37 | 21.7027027027 | 1.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 471y² = 1. Its smallest solution in positive whole numbers is x = 7,838,695, y = 361,188.
√471 in geometry and everyday measurements
- A square garage floor of 471 square feet measures about 21.7 ft (21 ft 8 in) per side, and its corner-to-corner diagonal is √942 ≈ 30.7 ft.
- 471 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √471 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √471 as its space diagonal.
Square roots near √471 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √474 | √474 | 21.7715 | No |
- The cube root of 471 is about 7.780490.
- Squaring undoes the root: (√471)² = 471, while 471² = 221,841 — the number whose square root is 471.
Frequently asked questions
What is the square root of 471?
The square root of 471 is √471, about 21.7025344142. The negative root, −21.702534, also squares to 471.
Is the square root of 471 rational or irrational?
Irrational. 471 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 √471 be simplified?
No. 471 = 3 × 157 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √471 rounded to two decimal places?
√471 ≈ 21.70 to two decimal places (21.7 to one, 21.703 to three). Check: 21.70² = 470.89, close to 471.