Question

1. \\(\frac{10}{6 + 2i}\\)

Explanation:

Step1: Rationalize the denominator

To simplify the complex fraction \(\frac{10}{6 + 2i}\), we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(6 + 2i\) is \(6 - 2i\).
\[
\frac{10}{6 + 2i} \times \frac{6 - 2i}{6 - 2i}
\]

Step2: Multiply the numerators and denominators

Multiply the numerators: \(10\times(6 - 2i)=60 - 20i\)
Multiply the denominators: \((6 + 2i)\times(6 - 2i)=6^{2}-(2i)^{2}=36 - 4i^{2}\)
Since \(i^{2}=- 1\), the denominator becomes \(36-4\times(-1)=36 + 4 = 40\)

Step3: Simplify the fraction

Now we have \(\frac{60 - 20i}{40}\), we can split the fraction: \(\frac{60}{40}-\frac{20i}{40}=\frac{3}{2}-\frac{1}{2}i\)

Answer:

\(\frac{3}{2}-\frac{1}{2}i\)