√256 at a glance
- Exact value
- 16
- Decimal (10 places)
- 16
- Rounded
- 16
- Perfect square?
- Yes — 16²
- Rational?
- Rational (whole number)
- Both square roots
- ±16
- Prime factorization
- 2⁸
- Cube root
- 6.349604
How to simplify √256
Write 256 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√256 = 2⁴ = 16
256 has 5 factor pairs (1 × 256, 2 × 128, 4 × 64, 8 × 32, 16 × 16). The pair with two equal numbers, 16 × 16, 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 256 = 1 + 3 + 5 + 7 + … + 31, the first 16 odd numbers.
Where √256 sits between perfect squares
256 is 16². The perfect squares on either side are 225 (15²) and 289 (17²), so the gaps are 31 below and 33 above — consecutive squares always differ by consecutive odd numbers.
Finding √256 with the Babylonian method
Picture a rectangle with an area of 256 and one side x; the other side must be 256 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √256.
To show the method at work on a perfect square, start one above the answer, at 17:
| Step | Guess x | 256 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 17.0000000000 | 15.0588235294 | 16.0294117647 | 1 |
| 2 | 16.0294117647 | 15.9706422018 | 16.0000269833 | 4 |
| 3 | 16.0000269833 | 15.9999730168 | 16.0000000000 | 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 √256 = 16.0000000000 to every decimal shown.
√256 in geometry and everyday measurements
- A square patio or deck of 256 square feet is exactly 16 ft (16 ft) on each side, so edging all the way around takes 4 × √256 ≈ 64 ft.
- 256 is not a sum of two whole-number squares — no pair of squares adds up to it — so √256 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √256 as its space diagonal.
Square roots near √256 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √253 | √253 | 15.9060 | No |
| √254 | √254 | 15.9374 | No |
| √255 | √255 | 15.9687 | No |
| √256 | 16 | 16.0000 | Yes |
| √257 | √257 | 16.0312 | No |
| √258 | √258 | 16.0624 | No |
| √259 | √259 | 16.0935 | No |
- The cube root of 256 is about 6.349604.
- Taking the square root twice gives the fourth root: √16 = 4, so 256 = 4⁴.
- Because 256 = 4 × 64, the root is twice √64: 2 × 8 ≈ 16.
Frequently asked questions
What is the square root of 256?
The square root of 256 is 16, because 16 × 16 = 256. Strictly, 256 has two square roots, 16 and −16; the √ symbol means the positive one.
Is the square root of 256 rational or irrational?
Rational. 256 is a perfect square, so its square root is the whole number 16.
Can √256 be simplified?
Yes, all the way to a whole number: √256 = 16.
What is √256 rounded to two decimal places?
√256 is exactly 16, or 16.00 written to two decimal places.