Question

1. \\(\frac{5i}{6 + 2i}\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(6 + 2i\) is \(6 - 2i\). So we have:
\[
\frac{5i}{6 + 2i} \times \frac{6 - 2i}{6 - 2i}
\]

Step2: Multiply the numerators.

Using the distributive property (FOIL for complex numbers), \(5i\times(6 - 2i)=5i\times6-5i\times2i = 30i-10i^{2}\). Since \(i^{2}=- 1\), this becomes \(30i - 10\times(-1)=30i + 10\).

Step3: Multiply the denominators.

Using the formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 6\) and \(b = 2i\), we get \(6^{2}-(2i)^{2}=36-4i^{2}\). Substituting \(i^{2}=-1\), we have \(36-4\times(-1)=36 + 4=40\).

Step4: Combine the results.

Now we have \(\frac{10 + 30i}{40}\). We can simplify this fraction by dividing each term in the numerator by 10: \(\frac{10}{40}+\frac{30i}{40}=\frac{1}{4}+\frac{3}{4}i\).

Answer:

\(\frac{1}{4}+\frac{3}{4}i\)