If `|((beta+gamma-alpha -delta)^4 , (beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4, (gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4, (alpha + beta-gamma-delta)^2,1)|=-k(alpha -beta)(alpha -gamma)(alpha-delta)(beta-gamma)(beta-delta)(gamma-delta)`, then the value of `(k)^(1//2)` is ____

Answer

To solve the given determinant problem, we will follow the steps outlined in the video transcript to arrive at the value of \( (k)^{1/2} \).

### Step-by-Step Solution:

1. **Define Variables**:
Let:
\[ x = \beta + \gamma - \alpha - \delta \]
\[ y = \gamma + \alpha - \beta - \delta \]
\[ z = \alpha + \beta - \gamma - \delta \]

2. **Set Up the Determinant**:
The determinant can be written as:
\[ \begin{vmatrix} x^4 & x^2 & 1 \\ y^4 & y^2 & 1 \\ z^4 & z^2 & 1 \end{vmatrix} \]

3. **Row Operations**:
Perform row operations to simplify the determinant:
- Subtract \( R_3 \) from \( R_1 \) and \( R_2 \):
\[ R_1 \rightarrow R_1 - R_3 \]
\[ R_2 \rightarrow R_2 - R_3 \]

4. **Factor Out Common Terms**:
After performing the row operations, factor out common terms:
\[ (x^2 - z^2) \quad \text{from } R_1 \]
\[ (y^2 - z^2) \quad \text{from } R_2 \]

5. **Rewrite the Determinant**:
The determinant now looks like:
\[ (x^2 - z^2)(y^2 - z^2) \begin{vmatrix} x^2 + z^2 & 1 & 0 \\ y^2 + z^2 & 1 & 0 \\ z^4 & z^2 & 1 \end{vmatrix} \]

6. **Expand the Determinant**:
Expand the determinant along the third column:
\[ = (x^2 - z^2)(y^2 - z^2) \cdot (x^2 - y^2) \]

7. **Calculate the Products**:
The product of the differences results in:
\[ (x^2 - z^2)(y^2 - z^2)(x^2 - y^2) \]

8. **Substitute Back**:
Substitute back the values of \( x, y, z \) to express in terms of \( \alpha, \beta, \gamma, \delta \):
\[ k = 64 \]

9. **Find \( (k)^{1/2} \)**:
Finally, compute:
\[ (k)^{1/2} = (64)^{1/2} = 8 \]

### Final Answer:
The value of \( (k)^{1/2} \) is \( 8 \).