Sure! Let's break down the given expression step by step to understand the calculation:
The given expression is:
[tex]\[
(-2)^4 + 5 \cdot \left\{[(-4) \cdot (+8) - (-7)] - \sqrt[3]{(-4)^3} \right\} - (-2)^{2^2}
\][/tex]
### Step 1: Calculating [tex]\((-2)^4\)[/tex]
[tex]\[
(-2)^4 = 16
\][/tex]
### Step 2: Calculating [tex]\((-2)^{2^2}\)[/tex]
First, we calculate the exponent:
[tex]\[
2^2 = 4
\][/tex]
Then:
[tex]\[
(-2)^4 = 16
\][/tex]
### Step 3: Calculating [tex]\((-4) \cdot (+8)\)[/tex]
[tex]\[
(-4) \cdot (+8) = -32
\][/tex]
### Step 4: Subtracting [tex]\(-(-7)\)[/tex] from the result of Step 3
[tex]\[
-32 - (-7) = -32 + 7 = -25
\][/tex]
### Step 5: Calculating [tex]\(\sqrt[3]{(-4)^3}\)[/tex]
[tex]\[
(-4)^3 = -64
\][/tex]
Taking the cube root of [tex]\(-64\)[/tex]:
[tex]\[
\sqrt[3]{(-64)} = (2 + 3.464101615137754j)
\][/tex]
(Note: This result includes a complex number since the cube root of a negative number can be complex.)
### Step 6: Subtracting the result from Step 5 from the result of Step 4
[tex]\[
-25 - (2 + 3.464101615137754j) = -27 - 3.464101615137754j
\][/tex]
### Step 7: Multiplying the result of Step 6 by 5
[tex]\[
5 \cdot (-27 - 3.464101615137754j) = -135 - 17.32050807568877j
\][/tex]
### Step 8: Adding the result of Step 1 to the result of Step 7
[tex]\[
16 + (-135 - 17.32050807568877j) = -119 - 17.32050807568877j
\][/tex]
### Step 9: Finally, subtracting the result of Step 2
[tex]\[
-119 - 17.32050807568877j - 16 = -135 - 17.32050807568877j
\][/tex]
Therefore, the result of the given expression is:
[tex]\[
\boxed{-135 - 17.32050807568877j}
\][/tex]
