To solve the problem, we need to simplify the given complex fraction:
\[
\frac{3 - 5i}{8 + 2i}
\]
Step 1: Multiply by the Conjugate of the Denominator
To eliminate the imaginary part in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(8 + 2i\) is \(8 - 2i\).
\[
\frac{3 - 5i}{8 + 2i} \cdot \frac{8 - 2i}{8 - 2i} = \frac{(3 - 5i)(8 - 2i)}{(8 + 2i)(8 - 2i)}
\]
Step 2: Simplify the Denominator
The denominator is a difference of squares:
\[
(8 + 2i)(8 - 2i) = 8^2 - (2i)^2 = 64 - 4i^2
\]
Since \(i^2 = -1\):
\[
64 - 4(-1) = 64 + 4 = 68
\]
So, the denominator simplifies to:
\[
68
\]
Step 3: Expand the Numerator
Now, expand the numerator \((3 - 5i)(8 - 2i)\) using the distributive property (FOIL method):
\[
(3 - 5i)(8 - 2i) = 3 \cdot 8 + 3 \cdot (-2i) + (-5i) \cdot 8 + (-5i) \cdot (-2i)
\]
Calculate each term:
\[
3 \cdot 8 = 24
\]
\[
3 \cdot (-2i) = -6i
\]
\[
(-5i) \cdot 8 = -40i
\]
\[
(-5i) \cdot (-2i) = 10i^2
\]
Since \(i^2 = -1\):
\[
10i^2 = 10(-1) = -10
\]
Combine all the terms:
\[
24 - 6i - 40i - 10 = 24 - 10 - 6i - 40i = 14 - 46i
\]
So, the numerator simplifies to:
\[
14 - 46i
\]
Step 4: Write the Simplified Fraction
Now, put the simplified numerator over the simplified denominator:
\[
\frac{14 - 46i}{68}
\]
Step 5: Separate into Real and Imaginary Parts
Separate the real and imaginary parts:
\[
\frac{14 - 46i}{68} = \frac{14}{68} - \frac{46i}{68}
\]
Simplify each fraction:
\[
\frac{14}{68} = \frac{7}{34}
\]
\[
\frac{46}{68} = \frac{23}{34}
\]
So, the expression becomes:
\[
\frac{7}{34} - \frac{23i}{34}
\]
Step 6: Match with the Given Options
The simplified form \(\frac{7}{34} - \frac{23i}{34}\) matches option C.
Final Answer:
\[
\boxed{C}
\]
Parent Tip: Review the logic above to help your child master the concept of free sat math worksheet.
