Let's solve the given expression step by step.
### Step-by-Step Solution:
#### Part 1: Solving the complex-number part
The expression we need to evaluate is:
[tex]\[ 23 - \left( 14 + i \left( 2 + (5 + 8 - 12)i \right) \right)^3 \][/tex]
1. Compute the inner expression with real numbers:
- [tex]\(5 + 8 - 12\)[/tex]:
[tex]\[ 5 + 8 - 12 = 1 \][/tex]
2. Evaluate the new inner imaginary expression:
- [tex]\(2 + (5 + 8 - 12)i\)[/tex]:
[tex]\[ 2 + (1)i = 2 + 1i \][/tex]
3. Substitute this result back into the main expression:
- [tex]\(14 + 1 \cdot (2 + 1i)\)[/tex]:
[tex]\[ 14 + 2 + 1i = 16 + 1i \][/tex]
4. Compute the cube of this complex number:
- [tex]\((16 + 1i)^3\)[/tex]:
[tex]\[ (16 + 1i)^3 = 4048 + 767i \][/tex]
5. Subtract this result from 23:
- [tex]\(23 - (16 + 1i)^3\)[/tex]:
[tex]\[ 23 - (4048 + 767i) = 23 - 4048 - 767i \][/tex]
[tex]\[ = -4025 - 767i \][/tex]
So, the result of the first part is:
[tex]\[ -4025 - 767i \][/tex]
#### Part 2: Solving the simpler real-number part
The expression we need to evaluate is:
[tex]\[ 100 - [14 + \{3 + 2(5 - 6 + 9)\} \cdot 1] \][/tex]
1. Compute the inner expression with real numbers:
- [tex]\(5 - 6 + 9\)[/tex]:
[tex]\[ 5 - 6 + 9 = 8 \][/tex]
2. Evaluate the new inner expression:
- [tex]\(3 + 2(5 - 6 + 9)\)[/tex]:
[tex]\[ 3 + 2 \cdot 8 = 3 + 16 = 19 \][/tex]
3. Substitute this result back into the main expression:
- [tex]\(14 + 19 \cdot 1\)[/tex]:
[tex]\[ 14 + 19 = 33 \][/tex]
4. Subtract this result from 100:
- [tex]\(100 - 33\)[/tex]:
[tex]\[ 100 - 33 = 67 \][/tex]
So, the result of the second part is:
[tex]\[ 67 \][/tex]
### Final Results:
- Part 1: [tex]\( 23 - \left( 14 + i(2 + (5+8-12)i) \right)^3 = -4025 - 767i \)[/tex]
- Part 2: [tex]\( 100 - [14 + \{3 + 2(5 - 6 + 9)\} \cdot 1] = 67 \)[/tex]
