Simplifying the Complex Expression \(\frac{{2 + 3i}}{{1 – 4i}}\)

Step 1: Identify the Expression

\(\frac{{2 + 3i}}{{1 – 4i}}\) – Comment: We start with the given expression and aim to simplify it.

Step 2: Multiply by the Conjugate

\(\frac{{2 + 3i}}{{1 – 4i}} \times \frac{{1 + 4i}}{{1 + 4i}}\) – Comment: We multiply by the conjugate to eliminate the imaginary part in the denominator.

Step 3: Expand the Numerator

\((2 + 3i)(1 + 4i) = 2 \cdot 1 + 2 \cdot 4i + 3i \cdot 1 + 3i \cdot 4i\) – Comment: Multiply real and imaginary parts separately.

\(= 2 + 8i + 3i + 12i^2\) – Comment: Combine like terms.

\(= 2 + 11i – 12\) – Comment: Use \(i^2 = -1\).

\(= -10 + 11i\) – Comment: Simplify to final form of the numerator.

Step 4: Expand the Denominator

\((1 – 4i)(1 + 4i) = 1^2 – (4i)^2\) – Comment: Apply the difference of squares formula.

\(= 1 – 16i^2\) – Comment: Use \(i^2 = -1\).

\(= 1 + 16\) – Comment: Simplify further.

\(= 17\) – Comment: Final form of the denominator.

Step 5: Divide Numerator by Denominator

\(\frac{{-10 + 11i}}{{17}} = \frac{{-10}}{{17}} + \frac{{11i}}{{17}}\) – Comment: Divide the numerator by the denominator.

\(\approx -0.5882 + 0.6471i\) – Comment: Approximate the result to a standard form.

Final Result: \(\frac{{2 + 3i}}{{1 – 4i}} \approx -0.5882 + 0.6471i\) – Comment: The final result represents the original expression in a simplified form.