A rectangular piece of land is to be sold off in Smaller pieces. The total area of the land is `2^(17)` sq. miles. The pieces to be cut out as `16^(2)sq`. Miles in size. How many smaller pieces of the land can be sold at the given size?

Options

A. `2^(15)`
B. `16^(4)`
C. `2^(9)`
D. None of these

Answer

To solve the problem, we need to determine how many smaller pieces of land can be sold from a rectangular piece of land with a total area of \(2^{17}\) square miles, when each smaller piece has an area of \(16^2\) square miles.

### Step-by-step Solution:

1. **Identify the total area of the land**:
The total area of the land is given as \(2^{17}\) square miles.

2. **Calculate the area of one smaller piece**:
The area of one smaller piece is given as \(16^2\) square miles.
We can express \(16\) in terms of base \(2\):
\[ 16 = 2^4 \]
Therefore,
\[ 16^2 = (2^4)^2 = 2^{4 \times 2} = 2^8 \]

3. **Set up the division to find the number of pieces**:
The number of smaller pieces that can be sold is equal to the total area divided by the area of one piece:
\[ \text{Number of pieces} = \frac{\text{Total Area}}{\text{Area of one piece}} = \frac{2^{17}}{2^8} \]

4. **Apply the laws of exponents**:
When dividing powers with the same base, we subtract the exponents:
\[ \frac{2^{17}}{2^8} = 2^{17 - 8} = 2^9 \]

5. **Calculate \(2^9\)**:
The value of \(2^9\) is:
\[ 2^9 = 512 \]

Thus, the number of smaller pieces of land that can be sold is **512**.