√484 at a glance
- Exact value
- 22
- Decimal (10 places)
- 22
- Rounded
- 22
- Perfect square?
- Yes — 22²
- Rational?
- Rational (whole number)
- Both square roots
- ±22
- Prime factorization
- 2² × 11²
- Cube root
- 7.851424
How to simplify √484
Write 484 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√484 = 2 × 11 = 22
484 has 5 factor pairs (1 × 484, 2 × 242, 4 × 121, 11 × 44, 22 × 22). The pair with two equal numbers, 22 × 22, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 484 = 1 + 3 + 5 + 7 + … + 43, the first 22 odd numbers.
Where √484 sits between perfect squares
484 is 22². The perfect squares on either side are 441 (21²) and 529 (23²), so the gaps are 43 below and 45 above — consecutive squares always differ by consecutive odd numbers.
Finding √484 with the Babylonian method
Picture a rectangle with an area of 484 and one side x; the other side must be 484 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √484.
To show the method at work on a perfect square, start one above the answer, at 23:
| Step | Guess x | 484 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 21.0434782609 | 22.0217391304 | 1 |
| 2 | 22.0217391304 | 21.9782823297 | 22.0000107301 | 4 |
| 3 | 22.0000107301 | 21.9999892699 | 22.0000000000 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √484 = 22.0000000000 to every decimal shown.
√484 in geometry and everyday measurements
- A square garage floor of 484 square feet measures exactly 22 ft (22 ft) per side, and its corner-to-corner diagonal is √968 ≈ 31.1 ft.
- 484 is not a sum of two whole-number squares — no pair of squares adds up to it — so √484 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 12 × 18 box, because 4² + 12² + 18² = 484.
Square roots near √484 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √481 | √481 | 21.9317 | No |
| √482 | √482 | 21.9545 | No |
| √483 | √483 | 21.9773 | No |
| √484 | 22 | 22.0000 | Yes |
| √485 | √485 | 22.0227 | No |
| √486 | 9√6 | 22.0454 | No |
| √487 | √487 | 22.0681 | No |
- The cube root of 484 is about 7.851424.
- Taking the square root twice gives the fourth root: √22 ≈ 4.690416.
- Because 484 = 4 × 121, the root is twice √121: 2 × 11 ≈ 22.
Frequently asked questions
What is the square root of 484?
The square root of 484 is 22, because 22 × 22 = 484. Strictly, 484 has two square roots, 22 and −22; the √ symbol means the positive one.
Is the square root of 484 rational or irrational?
Rational. 484 is a perfect square, so its square root is the whole number 22.
Can √484 be simplified?
Yes, all the way to a whole number: √484 = 22.
What is √484 rounded to two decimal places?
√484 is exactly 22, or 22.00 written to two decimal places.