Let us work through the expression step-by-step:
[tex]\[ = \sqrt[5]{-2 \cdot 2 - 8} - 2 + 4 \][/tex]
### Step 1: Calculate the expression inside the fifth root
First, we need to simplify the inner expression within the fifth root:
[tex]\[ -2 \cdot 2 - 8 \][/tex]
Perform the multiplication:
[tex]\[ -2 \cdot 2 = -4 \][/tex]
Next, subtract 8 from the result:
[tex]\[ -4 - 8 = -12 \][/tex]
So, the expression simplifies to:
[tex]\[ = \sqrt[5]{-12} - 2 + 4 \][/tex]
### Step 2: Take the fifth root of [tex]\(-12\)[/tex]
Now, we need to find the fifth root of [tex]\(-12\)[/tex]. The fifth root of a negative number is complex, and it can be expressed in the form of a complex number. By working this out, we get:
[tex]\[ \sqrt[5]{-12} = 1.329823164614347 + 0.9661730838189968 i \][/tex]
### Step 3: Perform the remaining arithmetic
Next, we perform the subtraction and addition outside of the fifth root:
[tex]\[ \left(1.329823164614347 + 0.9661730838189968 i\right) - 2 + 4 \][/tex]
First, subtract 2 from the real part (since 2 does not affect the imaginary part):
[tex]\[ 1.329823164614347 - 2 = -0.6701768353856528 \][/tex]
Now, add 4 to the real part:
[tex]\[ -0.6701768353856528 + 4 = 3.3298231646143472 \][/tex]
Finally, combine the real part with the unchanged imaginary part:
[tex]\[ 3.3298231646143472 + 0.9661730838189968 i \][/tex]
### Final Answer
Thus, the value of [tex]\( \sqrt[5]{-2 \cdot 2 - 8} - 2 + 4 \)[/tex] is:
[tex]\[ 3.3298231646143472 + 0.9661730838189968 i \][/tex]
