√468 at a glance
- Exact value
- 6√13
- Decimal (10 places)
- 21.6333076528
- Rounded
- 21.6 · 21.63 · 21.633
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.633308
- Prime factorization
- 2² × 3² × 13
- Cube root
- 7.763936
How to simplify √468
Look for the largest perfect square that divides 468. Here it is 36 (6²), because 468 = 36 × 13 and 13 has no square factor left:
The prime factorization tells the same story: 468 = 2² × 3² × 13. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 13 stays inside.
468 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √468 = 2√117, and √117 can be simplified again. Using 36 straight away finishes in one step.
Check: (6√13)² = 6² × 13 = 36 × 13 = 468. As a decimal, 6√13 = 6 × 3.6055512755 ≈ 21.6333076528.
Where √468 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √468 lies between 21 and 22. 468 is 27 above 441 and 16 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.6279 (0.02% low)
- Tangent from 21, i.e. 21 + 27 ÷ 42: 21.6429 (0.04% high)
- Tangent from 22, i.e. 22 − 16 ÷ 44: 21.6364 (0.01% high)
For √468 the tangent at 22 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 468 is just 16 below 484.
Finding √468 with the Babylonian method
Picture a rectangle with an area of 468 and one side x; the other side must be 468 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √468.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 468 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.2727272727 | 21.6363636364 | 2 |
| 2 | 21.6363636364 | 21.6302521008 | 21.6333078686 | 6 |
| 3 | 21.6333078686 | 21.6333074370 | 21.6333076528 | 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 √468 = 21.6333076528 to every decimal shown.
√468 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √468 the pattern is [21; 1, 1, 1, 2, 1, 1, 1, 42] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √468 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 | 6.3 × 10⁻¹ |
| 22/1 | 22.0000000000 | 3.7 × 10⁻¹ |
| 43/2 | 21.5000000000 | 1.3 × 10⁻¹ |
| 65/3 | 21.6666666667 | 3.3 × 10⁻² |
| 173/8 | 21.6250000000 | 8.3 × 10⁻³ |
| 238/11 | 21.6363636364 | 3.1 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 468y² = 1. Its smallest solution in positive whole numbers is x = 649, y = 30.
√468 in geometry and everyday measurements
- A square garage floor of 468 square feet measures about 21.63 ft (21 ft 8 in) per side, and its corner-to-corner diagonal is √936 ≈ 30.6 ft.
- 468 = 12² + 18², so by the Pythagorean theorem √468 is the diagonal of a 12 × 18 rectangle — and the distance between the points (0, 0) and (12, 18) on a grid.
- Since √468 = 6√13, a length of √468 is exactly 6 copies of the length √13 laid end to end.
Square roots near √468 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √465 | √465 | 21.5639 | No |
| √466 | √466 | 21.5870 | No |
| √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 |
- The cube root of 468 is about 7.763936.
- Because 468 = 4 × 117, the root is twice √117: 2 × 10.816654 ≈ 21.633308.
Frequently asked questions
What is the square root of 468?
The square root of 468 is 6√13 in simplest radical form, which is about 21.6333076528. The negative root, −21.633308, also squares to 468.
Is the square root of 468 rational or irrational?
Irrational. 468 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 √468 be simplified?
Yes. The largest perfect square dividing 468 is 36, so √468 = √36 × √13 = 6√13.
What is √468 rounded to two decimal places?
√468 ≈ 21.63 to two decimal places (21.6 to one, 21.633 to three). Check: 21.63² = 467.8569, close to 468.