To solve the equation \((\vec{a} \times \vec{b}) \cdot [(\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})] = (\vec{a} \cdot (\vec{b} \times \vec{c}))^k\) and find the value of \(k\), we can follow these steps:
### Step 1: Simplify the Left-Hand Side
We start with the left-hand side of the equation:
\[
(\vec{a} \times \vec{b}) \cdot [(\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})]
\]
Using the vector triple product identity, we know that:
\[
\vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z}
\]
Let \(\vec{x} = \vec{b}\), \(\vec{y} = \vec{c}\), and \(\vec{z} = \vec{a}\). Therefore:
\[
(\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}
\]
### Step 2: Substitute Back into the Equation
Now we substitute this back into our left-hand side:
\[
(\vec{a} \times \vec{b}) \cdot [(\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}]
\]
Distributing the dot product:
\[
(\vec{a} \times \vec{b}) \cdot ((\vec{b} \cdot \vec{a}) \vec{c}) - (\vec{a} \times \vec{b}) \cdot ((\vec{b} \cdot \vec{c}) \vec{a})
\]
### Step 3: Evaluate Each Dot Product
The first term:
\[
(\vec{a} \times \vec{b}) \cdot ((\vec{b} \cdot \vec{a}) \vec{c}) = (\vec{b} \cdot \vec{a}) ((\vec{a} \times \vec{b}) \cdot \vec{c})
\]
The second term:
\[
(\vec{a} \times \vec{b}) \cdot ((\vec{b} \cdot \vec{c}) \vec{a}) = (\vec{b} \cdot \vec{c}) ((\vec{a} \times \vec{b}) \cdot \vec{a}) = 0
\]
(since \((\vec{a} \times \vec{b}) \cdot \vec{a} = 0\)).
Thus, we have:
\[
(\vec{a} \times \vec{b}) \cdot [(\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})] = (\vec{b} \cdot \vec{a}) ((\vec{a} \times \vec{b}) \cdot \vec{c})
\]
### Step 4: Relate to the Right-Hand Side
Now, we need to relate this to the right-hand side:
\[
(\vec{a} \cdot (\vec{b} \times \vec{c}))^k
\]
Using the scalar triple product, we know:
\[
\vec{a} \cdot (\vec{b} \times \vec{c}) = \text{det}(\vec{a}, \vec{b}, \vec{c})
\]
Thus, we can express:
\[
(\vec{b} \cdot \vec{a}) ((\vec{a} \times \vec{b}) \cdot \vec{c}) = \text{det}(\vec{a}, \vec{b}, \vec{c})^k
\]
### Step 5: Determine the Value of \(k\)
From the left-hand side, we can see that the expression simplifies to:
\[
\text{det}(\vec{a}, \vec{b}, \vec{c})^2
\]
Thus, we can equate:
\[
\text{det}(\vec{a}, \vec{b}, \vec{c})^2 = \text{det}(\vec{a}, \vec{b}, \vec{c})^k
\]
This implies that:
\[
k = 2
\]
### Final Answer
The value of \(k\) is:
\[
\boxed{2}
\]
