Certainly! Let's solve the given mathematical expression step-by-step:
1. Calculate [tex]\((-2)^4\)[/tex]:
[tex]\[
(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16
\][/tex]
2. Calculate [tex]\((-4) \cdot (+8)\)[/tex]:
[tex]\[
(-4) \times (+8) = -32
\][/tex]
3. Calculate the cube of [tex]\(-4\)[/tex], which is [tex]\((-4)^3\)[/tex]:
[tex]\[
(-4)^3 = (-4) \times (-4) \times (-4) = -64
\][/tex]
4. Calculate the cube root of [tex]\((-4)^3\)[/tex], which is [tex]\(\sqrt[3]{(-4)^3}\)[/tex]:
[tex]\[
\sqrt[3]{(-4)^3} = \sqrt[3]{-64} = 2 + 3.464101615137754j
\][/tex]
Given that the cube root of a negative number is a complex number in addition to real parts, in this case, this is [tex]\(2 + 3.464101615137754j\)[/tex].
5. Calculate the expression inside the curly braces:
[tex]\[
(-32 - (-7)) - \sqrt[3]{-64} = (-32 + 7) - (2 + 3.464101615137754j) = -25 - (2 + 3.464101615137754j) = -27 - 3.464101615137754j
\][/tex]
6. Multiply by 5:
[tex]\[
5 \cdot (-27 - 3.464101615137754j) = -135 - 17.32050807568877j
\][/tex]
7. Calculate [tex]\((-2)^2\)[/tex]:
[tex]\[
(-2)^2 = (-2) \times (-2) = 4
\][/tex]
8. Combine all parts to complete the final expression:
[tex]\[
16 + (-135 - 17.32050807568877j) - 4 = 16 - 135 - 17.32050807568877j - 4 = -123 - 17.32050807568877j
\][/tex]
So, the final solution to the given expression is:
[tex]\[
\boxed{-123 - 17.32050807568877j}
\][/tex]
