√473 at a glance
- Exact value
- √473
- Decimal (10 places)
- 21.7485631709
- Rounded
- 21.7 · 21.75 · 21.749
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.748563
- Prime factorization
- 11 × 43
- Cube root
- 7.791488
How to simplify √473
The prime factorization of 473 is 11 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √473 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 473, 11 and 43 appear an odd number of times, so √473 is irrational and 21.7485631709 is a rounded value.
Where √473 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √473 lies between 21 and 22. 473 is 32 above 441 and 11 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.7442 (0.02% low)
- Tangent from 21, i.e. 21 + 32 ÷ 42: 21.7619 (0.06% high)
- Tangent from 22, i.e. 22 − 11 ÷ 44: 21.7500 (0.01% high)
For √473 the tangent at 22 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 473 is just 11 below 484.
Finding √473 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 473: following the tangent line down to zero simplifies to averaging x with 473 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 473 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.5000000000 | 21.7500000000 | 2 |
| 2 | 21.7500000000 | 21.7471264368 | 21.7485632184 | 7 |
| 3 | 21.7485632184 | 21.7485631235 | 21.7485631709 | 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 √473 = 21.7485631709 to every decimal shown.
√473 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √473 the pattern is [21; 1, 2, 1, 42] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √473 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 | 7.5 × 10⁻¹ |
| 22/1 | 22.0000000000 | 2.5 × 10⁻¹ |
| 65/3 | 21.6666666667 | 8.2 × 10⁻² |
| 87/4 | 21.7500000000 | 1.4 × 10⁻³ |
| 3,719/171 | 21.7485380117 | 2.5 × 10⁻⁵ |
| 3,806/175 | 21.7485714286 | 8.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 473y² = 1. Its smallest solution in positive whole numbers is x = 87, y = 4.
√473 in geometry and everyday measurements
- A square garage floor of 473 square feet measures about 21.75 ft (21 ft 9 in) per side, and its corner-to-corner diagonal is √946 ≈ 30.8 ft.
- 473 is not a sum of two whole-number squares — the prime factor 11 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √473 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 8 × 20 box, because 3² + 8² + 20² = 473.
Square roots near √473 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √475 | 5√19 | 21.7945 | No |
| √476 | 2√119 | 21.8174 | No |
- The cube root of 473 is about 7.791488.
- Squaring undoes the root: (√473)² = 473, while 473² = 223,729 — the number whose square root is 473.
Frequently asked questions
What is the square root of 473?
The square root of 473 is √473, about 21.7485631709. The negative root, −21.748563, also squares to 473.
Is the square root of 473 rational or irrational?
Irrational. 473 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 √473 be simplified?
No. 473 = 11 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √473 rounded to two decimal places?
√473 ≈ 21.75 to two decimal places (21.7 to one, 21.749 to three). Check: 21.75² = 473.0625, close to 473.