√784 at a glance
- Exact value
- 28
- Decimal (10 places)
- 28
- Rounded
- 28
- Perfect square?
- Yes — 28²
- Rational?
- Rational (whole number)
- Both square roots
- ±28
- Prime factorization
- 2⁴ × 7²
- Cube root
- 9.220873
How to simplify √784
Write 784 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√784 = 2² × 7 = 28
784 has 8 factor pairs (1 × 784, 2 × 392, 4 × 196, 7 × 112, 8 × 98, 14 × 56, 16 × 49, 28 × 28). The pair with two equal numbers, 28 × 28, 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 784 = 1 + 3 + 5 + 7 + … + 55, the first 28 odd numbers.
Where √784 sits between perfect squares
784 is 28². The perfect squares on either side are 729 (27²) and 841 (29²), so the gaps are 55 below and 57 above — consecutive squares always differ by consecutive odd numbers.
Finding √784 with the Babylonian method
Picture a rectangle with an area of 784 and one side x; the other side must be 784 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √784.
To show the method at work on a perfect square, start one above the answer, at 29:
| Step | Guess x | 784 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 27.0344827586 | 28.0172413793 | 1 |
| 2 | 28.0172413793 | 27.9827692308 | 28.0000053050 | 5 |
| 3 | 28.0000053050 | 27.9999946950 | 28.0000000000 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √784 = 28.0000000000 to every decimal shown.
√784 in geometry and everyday measurements
- 784 square feet is 72.8 m². Laid out as a square — a small house footprint or a lot — it is exactly 28 ft (28 ft) on a side.
- 784 is not a sum of two whole-number squares — no pair of squares adds up to it — so √784 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 12 × 24 box, because 8² + 12² + 24² = 784.
Square roots near √784 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √781 | √781 | 27.9464 | No |
| √782 | √782 | 27.9643 | No |
| √783 | 3√87 | 27.9821 | No |
| √784 | 28 | 28.0000 | Yes |
| √785 | √785 | 28.0179 | No |
| √786 | √786 | 28.0357 | No |
| √787 | √787 | 28.0535 | No |
- The cube root of 784 is about 9.220873.
- Taking the square root twice gives the fourth root: √28 ≈ 5.291503.
- Because 784 = 4 × 196, the root is twice √196: 2 × 14 ≈ 28.
Frequently asked questions
What is the square root of 784?
The square root of 784 is 28, because 28 × 28 = 784. Strictly, 784 has two square roots, 28 and −28; the √ symbol means the positive one.
Is the square root of 784 rational or irrational?
Rational. 784 is a perfect square, so its square root is the whole number 28.
Can √784 be simplified?
Yes, all the way to a whole number: √784 = 28.
What is √784 rounded to two decimal places?
√784 is exactly 28, or 28.00 written to two decimal places.