Find the values of the following determinants
`|{:(4,0,2),(1,5,-6),(3,-2,8):}|`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 4 & 0 & 2 \\ 1 & 5 & -6 \\ 3 & -2 & 8 \end{vmatrix} \]

we can use the method of cofactor expansion along the first row.

### Step 1: Write the determinant and identify the elements of the first row.
The first row of the determinant is \(4, 0, 2\).

### Step 2: Expand the determinant along the first row.
Using cofactor expansion, we have:

\[ D = 4 \cdot \begin{vmatrix} 5 & -6 \\ -2 & 8 \end{vmatrix} - 0 \cdot \begin{vmatrix} 1 & -6 \\ 3 & 8 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & 5 \\ 3 & -2 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants.
1. For the first determinant:

\[ \begin{vmatrix} 5 & -6 \\ -2 & 8 \end{vmatrix} = (5 \cdot 8) - (-6 \cdot -2) = 40 - 12 = 28 \]

2. The second determinant is multiplied by 0, so it contributes nothing.

3. For the third determinant:

\[ \begin{vmatrix} 1 & 5 \\ 3 & -2 \end{vmatrix} = (1 \cdot -2) - (5 \cdot 3) = -2 - 15 = -17 \]

### Step 4: Substitute back into the determinant equation.
Now substituting back, we get:

\[ D = 4 \cdot 28 + 2 \cdot (-17) \]

### Step 5: Simplify the expression.
Calculating this gives:

\[ D = 112 - 34 = 78 \]

### Final Result:
Thus, the value of the determinant is

\[ \boxed{78} \]