√450 at a glance
- Exact value
- 15√2
- Decimal (10 places)
- 21.2132034356
- Rounded
- 21.2 · 21.21 · 21.213
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.213203
- Prime factorization
- 2 × 3² × 5²
- Cube root
- 7.663094
How to simplify √450
Look for the largest perfect square that divides 450. Here it is 225 (15²), because 450 = 225 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 450 = 2 × 3² × 5². Each pair of equal primes leaves the radical as one factor, so 3 × 5 comes out and 2 stays inside.
450 has 3 square factors (9, 25 and 225). Starting with a smaller one still works but takes more rounds: √450 = 3√50, and √50 can be simplified again. Using 225 straight away finishes in one step.
Check: (15√2)² = 15² × 2 = 225 × 2 = 450. As a decimal, 15√2 = 15 × 1.4142135624 ≈ 21.2132034356.
Where √450 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √450 lies between 21 and 22. 450 is 9 above 441 and 34 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.2093 (0.02% low)
- Tangent from 21, i.e. 21 + 9 ÷ 42: 21.2143 (0.01% high)
- Tangent from 22, i.e. 22 − 34 ÷ 44: 21.2273 (0.07% high)
For √450 the tangent at 21 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 450 is just 9 above 441.
Finding √450 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, 21 (21² = 441):
| Step | Guess x | 450 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.4285714286 | 21.2142857143 | 2 |
| 2 | 21.2142857143 | 21.2121212121 | 21.2132034632 | 7 |
| 3 | 21.2132034632 | 21.2132034080 | 21.2132034356 | 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 √450 = 21.2132034356 to every decimal shown.
√450 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √450 the pattern is [21; 4, 1, 2, 4, 2, 1, 4, 42] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √450 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 | 2.1 × 10⁻¹ |
| 85/4 | 21.2500000000 | 3.7 × 10⁻² |
| 106/5 | 21.2000000000 | 1.3 × 10⁻² |
| 297/14 | 21.2142857143 | 1.1 × 10⁻³ |
| 1,294/61 | 21.2131147541 | 8.9 × 10⁻⁵ |
| 2,885/136 | 21.2132352941 | 3.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 450y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 924.
√450 in geometry and everyday measurements
- A square garage floor of 450 square feet measures about 21.21 ft (21 ft 3 in) per side, and its corner-to-corner diagonal is √900 ≈ 30 ft.
- 450 = 3² + 21² = 15² + 15², so by the Pythagorean theorem √450 is the diagonal of rectangles measuring 3 × 21 and 15 × 15 — and the distance between the points (0, 0) and (3, 21) on a grid.
- Since √450 = 15√2, a length of √450 is exactly 15 copies of the length √2 laid end to end.
Square roots near √450 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √447 | √447 | 21.1424 | No |
| √448 | 8√7 | 21.1660 | No |
| √449 | √449 | 21.1896 | No |
| √450 | 15√2 | 21.2132 | No |
| √451 | √451 | 21.2368 | No |
| √452 | 2√113 | 21.2603 | No |
| √453 | √453 | 21.2838 | No |
- The cube root of 450 is about 7.663094.
- Squaring undoes the root: (√450)² = 450, while 450² = 202,500 — the number whose square root is 450.
Frequently asked questions
What is the square root of 450?
The square root of 450 is 15√2 in simplest radical form, which is about 21.2132034356. The negative root, −21.213203, also squares to 450.
Is the square root of 450 rational or irrational?
Irrational. 450 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 √450 be simplified?
Yes. The largest perfect square dividing 450 is 225, so √450 = √225 × √2 = 15√2.
What is √450 rounded to two decimal places?
√450 ≈ 21.21 to two decimal places (21.2 to one, 21.213 to three). Check: 21.21² = 449.8641, close to 450.