The given equation is:
[tex]\frac{5+3i}{1-4i}[/tex]A) Complex conjuagte of the denominator = 1 + 41
The complex conjugate of an expression is the opposite of the sign between the real and imagiary parts
The denomiator in this case is 1 - 4i, the complex conjugate is therefore 1 + 4i
B) The mistakes made by Melissa:
Mistake 1: Melissa said i² = -i, this is wrong. i² = -1
Mistake 2: In the numerator, (5+3i)(1+4i) gave Melissa 5 + 12i², this is wrong.
(5+3i)(1+4i) = 5 + 20i + 3i + 12i² = 5 + 23i + 12(-1) = 5 - 12 + 23i
(5+3i)(1+4i) = -7 + 23i
Mistake 3: In the denominator, (1 - 4i)(1+4i) gave Melissa 16i + 1, this is wrong.
(1 - 4i)(1+4i) = 1 + 4i - 4i - 16i² = 1 - 16i² = 1 - 16(-1) = 1 + 16
(1 - 4i)(1+4i) = 17
c) The correct solution of the problem:
[tex]\frac{5+3i}{1-4i}[/tex]Step 1: Rationalise, that is multiply the numerator and the denominator by the conjugate of 1 - 4i, the conjugate is 1 + 4i
[tex]\begin{gathered} \frac{5+3i}{1-4i}\times\frac{1+4i}{1+4i} \\ \text{Step}2\colon\text{ }\frac{5+20i+3i+12i^2}{1+4i-4i-16i^2} \\ \text{Step}3\colon\text{ }\frac{5+23i+12(-1)}{1-16(-1)} \\ \text{Step}4\colon\text{ }\frac{-7+23i}{17} \\ \text{Step}5\colon\text{ }\frac{-7}{17}+\frac{23}{17}i \end{gathered}[/tex][tex]\text{Answer: }\frac{-7}{17}+\frac{23}{17}i[/tex]D) Real part = -7/17
Imaginary part = 23/17
