√474 at a glance
- Exact value
- √474
- Decimal (10 places)
- 21.7715410571
- Rounded
- 21.8 · 21.77 · 21.772
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.771541
- Prime factorization
- 2 × 3 × 79
- Cube root
- 7.796975
How to simplify √474
The prime factorization of 474 is 2 × 3 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √474 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 474, 2, 3 and 79 appear an odd number of times, so √474 is irrational and 21.7715410571 is a rounded value.
Where √474 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √474 lies between 21 and 22. 474 is 33 above 441 and 10 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.7674 (0.02% low)
- Tangent from 21, i.e. 21 + 33 ÷ 42: 21.7857 (0.07% high)
- Tangent from 22, i.e. 22 − 10 ÷ 44: 21.7727 (0.01% high)
For √474 the tangent at 22 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 474 is just 10 below 484.
Finding √474 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 | 474 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.5454545455 | 21.7727272727 | 2 |
| 2 | 21.7727272727 | 21.7703549061 | 21.7715410894 | 7 |
| 3 | 21.7715410894 | 21.7715410248 | 21.7715410571 | 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 √474 = 21.7715410571 to every decimal shown.
√474 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √474 the pattern is [21; 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, …] 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 √474 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 | 7.7 × 10⁻¹ |
| 22/1 | 22.0000000000 | 2.3 × 10⁻¹ |
| 87/4 | 21.7500000000 | 2.2 × 10⁻² |
| 196/9 | 21.7777777778 | 6.2 × 10⁻³ |
| 283/13 | 21.7692307692 | 2.3 × 10⁻³ |
| 479/22 | 21.7727272727 | 1.2 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 474y² = 1. Its smallest solution in positive whole numbers is x = 193,549, y = 8,890.
√474 in geometry and everyday measurements
- A square garage floor of 474 square feet measures about 21.77 ft (21 ft 9 in) per side, and its corner-to-corner diagonal is √948 ≈ 30.8 ft.
- 474 is not a sum of two whole-number squares — the prime factor 3 and 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √474 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 13 × 17 box, because 4² + 13² + 17² = 474.
Square roots near √474 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √477 | 3√53 | 21.8403 | No |
- The cube root of 474 is about 7.796975.
- Squaring undoes the root: (√474)² = 474, while 474² = 224,676 — the number whose square root is 474.
Frequently asked questions
What is the square root of 474?
The square root of 474 is √474, about 21.7715410571. The negative root, −21.771541, also squares to 474.
Is the square root of 474 rational or irrational?
Irrational. 474 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 √474 be simplified?
No. 474 = 2 × 3 × 79 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √474 rounded to two decimal places?
√474 ≈ 21.77 to two decimal places (21.8 to one, 21.772 to three). Check: 21.77² = 473.9329, close to 474.