The value of `|{:(2^(2),2^(3),2^(4)),(2^(3),2^(4),2^(5)),(2^(4),2^(5),2^(6)):}|` is

Options

A. `2^(9)`
B. `2^(6)`
C. `2^(13)`
D. 0

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6 \end{vmatrix} \]

we can follow these steps:

### Step 1: Factor out common terms from each row
We can take out common factors from each row of the determinant.

- From the first row, we can take out \(2^2\).
- From the second row, we can take out \(2^3\).
- From the third row, we can take out \(2^4\).

This gives us:

\[ D = 2^2 \cdot 2^3 \cdot 2^4 \cdot \begin{vmatrix} 1 & 2 & 2^2 \\ 1 & 2 & 2^2 \\ 1 & 2 & 2^2 \end{vmatrix} \]

### Step 2: Simplify the determinant
Now, we can simplify the determinant:

\[ D = 2^{2+3+4} \cdot \begin{vmatrix} 1 & 2 & 2^2 \\ 1 & 2 & 2^2 \\ 1 & 2 & 2^2 \end{vmatrix} \]

Calculating \(2^{2+3+4}\):

\[ 2^{2+3+4} = 2^9 \]

### Step 3: Evaluate the determinant
Next, we notice that the rows of the matrix

\[ \begin{vmatrix} 1 & 2 & 2^2 \\ 1 & 2 & 2^2 \\ 1 & 2 & 2^2 \end{vmatrix} \]

are identical. Since two rows of a determinant are the same, the value of the determinant is 0.

### Step 4: Combine results
Thus, we have:

\[ D = 2^9 \cdot 0 = 0 \]

### Final Answer
The value of the determinant is

\[ \boxed{0} \]

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