Let's solve the given equation step by step. The given equation is:
[tex]\[
\frac{1}{x - y i} = (1 + 2i)^2 - \left( \frac{1 + 4i}{4 - i} \right)^3
\][/tex]
We'll start by examining both sides of the equation.
### Left Side of the Equation
The left side is given by:
[tex]\[
\frac{1}{x - y i}
\][/tex]
This term represents a complex fraction with [tex]\(i\)[/tex] representing the imaginary unit.
### Right Side of the Equation
Let’s break down the right side of the equation into separate parts and solve each part step by step.
1. First Term: [tex]\((1 + 2i)^2\)[/tex]
To compute [tex]\((1 + 2i)^2\)[/tex]:
[tex]\[
(1 + 2i)^2 = (1 + 2i)(1 + 2i)
\][/tex]
Applying the distributive property (FOIL method):
[tex]\[
(1 + 2i)(1 + 2i) = 1 \cdot 1 + 1 \cdot 2i + 2i \cdot 1 + 2i \cdot 2i
\][/tex]
[tex]\[
= 1 + 2i + 2i + 4i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 1 + 4i + 4(-1)
\][/tex]
[tex]\[
= 1 + 4i - 4
\][/tex]
[tex]\[
= -3 + 4i
\][/tex]
2. Second Term: [tex]\(\left( \frac{1 + 4i}{4 - i} \right)^3\)[/tex]
Let’s start by simplifying [tex]\(\frac{1 + 4i}{4 - i}\)[/tex]:
To do this, we will multiply by the conjugate of the denominator:
[tex]\[
\frac{1 + 4i}{4 - i} \cdot \frac{4 + i}{4 + i} = \frac{(1 + 4i)(4 + i)}{(4 - i)(4 + i)}
\][/tex]
Expanding both the numerator and the denominator:
The denominator:
[tex]\[
(4 - i)(4 + i) = 4^2 - i^2 = 16 - (-1) = 16 + 1 = 17
\][/tex]
The numerator:
[tex]\[
(1 + 4i)(4 + i) = 1 \cdot 4 + 1 \cdot i + 4i \cdot 4 + 4i \cdot i
\][/tex]
[tex]\[
= 4 + i + 16i + 4i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 4 + i + 16i + 4(-1)
\][/tex]
[tex]\[
= 4 + 17i - 4
\][/tex]
[tex]\[
= 17i
\][/tex]
Thus:
[tex]\[
\frac{1 + 4i}{4 - i} = \frac{17i}{17} = i
\][/tex]
Cubing this result:
[tex]\[
\left( i \right)^3 = -i
\][/tex]
Combining the results from both terms on the right side:
[tex]\[
(1 + 2i)^2 - \left( \frac{1 + 4i}{4 - i} \right)^3 = (-3 + 4i) - (-i)
\][/tex]
Simplifying further:
[tex]\[
= -3 + 4i + i = -3 + 5i
\][/tex]
Now we have:
[tex]\[
\frac{1}{x - y i} = -3 + 5i
\][/tex]
Simplifying the left side, we get:
[tex]\[
\frac{1}{x - y i}
\][/tex]
Hence, the solution in its simplified form is:
[tex]\[
\frac{1}{-i y + x} = -3 + 5i
\][/tex]
This completes the detailed step-by-step solution of the given equation.
