√477 at a glance
- Exact value
- 3√53
- Decimal (10 places)
- 21.8403296678
- Rounded
- 21.8 · 21.84 · 21.840
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.840330
- Prime factorization
- 3² × 53
- Cube root
- 7.813389
How to simplify √477
Look for the largest perfect square that divides 477. Here it is 9 (3²), because 477 = 9 × 53 and 53 has no square factor left:
The prime factorization tells the same story: 477 = 3² × 53. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 53 stays inside.
Check: (3√53)² = 3² × 53 = 9 × 53 = 477. As a decimal, 3√53 = 3 × 7.2801098893 ≈ 21.8403296678.
Where √477 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √477 lies between 21 and 22. 477 is 36 above 441 and 7 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.8372 (0.01% low)
- Tangent from 21, i.e. 21 + 36 ÷ 42: 21.8571 (0.08% high)
- Tangent from 22, i.e. 22 − 7 ÷ 44: 21.8409 (0% high)
For √477 the tangent at 22 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 477 is just 7 below 484.
Finding √477 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 477: following the tangent line down to zero simplifies to averaging x with 477 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 477 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.6818181818 | 21.8409090909 | 3 |
| 2 | 21.8409090909 | 21.8397502601 | 21.8403296755 | 8 |
| 3 | 21.8403296755 | 21.8403296602 | 21.8403296678 | 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 √477 = 21.8403296678 to every decimal shown.
√477 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √477 the pattern is [21; 1, 5, 3, 1, 4, 10, 1, 2, 2, 4, 2, 2, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √477 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 | 8.4 × 10⁻¹ |
| 22/1 | 22.0000000000 | 1.6 × 10⁻¹ |
| 131/6 | 21.8333333333 | 7.0 × 10⁻³ |
| 415/19 | 21.8421052632 | 1.8 × 10⁻³ |
| 546/25 | 21.8400000000 | 3.3 × 10⁻⁴ |
| 2,599/119 | 21.8403361345 | 6.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 477y² = 1. Its smallest solution in positive whole numbers is x = 8,777,860,001, y = 401,910,600.
√477 in geometry and everyday measurements
- A square garage floor of 477 square feet measures about 21.84 ft (21 ft 10 in) per side, and its corner-to-corner diagonal is √954 ≈ 30.9 ft.
- 477 = 6² + 21², so by the Pythagorean theorem √477 is the diagonal of a 6 × 21 rectangle — and the distance between the points (0, 0) and (6, 21) on a grid.
- Since √477 = 3√53, a length of √477 is exactly 3 copies of the length √53 laid end to end.
Square roots near √477 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √474 | √474 | 21.7715 | No |
| √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 |
- The cube root of 477 is about 7.813389.
- Squaring undoes the root: (√477)² = 477, while 477² = 227,529 — the number whose square root is 477.
Frequently asked questions
What is the square root of 477?
The square root of 477 is 3√53 in simplest radical form, which is about 21.8403296678. The negative root, −21.840330, also squares to 477.
Is the square root of 477 rational or irrational?
Irrational. 477 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 √477 be simplified?
Yes. The largest perfect square dividing 477 is 9, so √477 = √9 × √53 = 3√53.
What is √477 rounded to two decimal places?
√477 ≈ 21.84 to two decimal places (21.8 to one, 21.840 to three). Check: 21.84² = 476.9856, close to 477.