√500 at a glance
- Exact value
- 10√5
- Decimal (10 places)
- 22.3606797750
- Rounded
- 22.4 · 22.36 · 22.361
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.360680
- Prime factorization
- 2² × 5³
- Cube root
- 7.937005
How to simplify √500
Look for the largest perfect square that divides 500. Here it is 100 (10²), because 500 = 100 × 5 and 5 has no square factor left:
The prime factorization tells the same story: 500 = 2² × 5³. Each pair of equal primes leaves the radical as one factor, so 2 × 5 comes out and 5 stays inside.
500 has 3 square factors (4, 25 and 100). Starting with a smaller one still works but takes more rounds: √500 = 2√125, and √125 can be simplified again. Using 100 straight away finishes in one step.
Check: (10√5)² = 10² × 5 = 100 × 5 = 500. As a decimal, 10√5 = 10 × 2.2360679775 ≈ 22.3606797750.
Where √500 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √500 lies between 22 and 23. 500 is 16 above 484 and 29 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.3556 (0.02% low)
- Tangent from 22, i.e. 22 + 16 ÷ 44: 22.3636 (0.01% high)
- Tangent from 23, i.e. 23 − 29 ÷ 46: 22.3696 (0.04% high)
For √500 the tangent at 22 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 500 is just 16 above 484.
Finding √500 with the Babylonian method
Picture a rectangle with an area of 500 and one side x; the other side must be 500 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √500.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 500 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.7272727273 | 22.3636363636 | 2 |
| 2 | 22.3636363636 | 22.3577235772 | 22.3606799704 | 6 |
| 3 | 22.3606799704 | 22.3606795796 | 22.3606797750 | 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 √500 = 22.3606797750 to every decimal shown.
√500 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √500 the pattern is [22; 2, 1, 3, 2, 1, 1, 10, 1, 1, 2, 3, 1, …] 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 √500 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 3.6 × 10⁻¹ |
| 45/2 | 22.5000000000 | 1.4 × 10⁻¹ |
| 67/3 | 22.3333333333 | 2.7 × 10⁻² |
| 246/11 | 22.3636363636 | 3.0 × 10⁻³ |
| 559/25 | 22.3600000000 | 6.8 × 10⁻⁴ |
| 805/36 | 22.3611111111 | 4.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 500y² = 1. Its smallest solution in positive whole numbers is x = 930,249, y = 41,602.
√500 in geometry and everyday measurements
- A square garage floor of 500 square feet measures about 22.36 ft (22 ft 4 in) per side, and its corner-to-corner diagonal is √1000 ≈ 31.6 ft.
- 500 = 4² + 22² = 10² + 20², so by the Pythagorean theorem √500 is the diagonal of rectangles measuring 4 × 22 and 10 × 20 — and the distance between the points (0, 0) and (4, 22) on a grid.
- Since √500 = 10√5, a length of √500 is exactly 10 copies of the length √5 laid end to end.
Square roots near √500 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √497 | √497 | 22.2935 | No |
| √498 | √498 | 22.3159 | No |
| √499 | √499 | 22.3383 | No |
| √500 | 10√5 | 22.3607 | No |
| √501 | √501 | 22.3830 | No |
| √502 | √502 | 22.4054 | No |
| √503 | √503 | 22.4277 | No |
- The cube root of 500 is about 7.937005.
- Dividing by 100 divides the root by 10: √5 = √500 ÷ 10 ≈ 2.23606798.
Frequently asked questions
What is the square root of 500?
The square root of 500 is 10√5 in simplest radical form, which is about 22.3606797750. The negative root, −22.360680, also squares to 500.
Is the square root of 500 rational or irrational?
Irrational. 500 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √500 be simplified?
Yes. The largest perfect square dividing 500 is 100, so √500 = √100 × √5 = 10√5.
What is √500 rounded to two decimal places?
√500 ≈ 22.36 to two decimal places (22.4 to one, 22.361 to three). Check: 22.36² = 499.9696, close to 500.