Answer:
[tex]\frac{5i}{3 - 4i} = \frac{3i - 4}{5}[/tex]
Step-by-step explanation:
Given
[tex]\frac{5i}{3 - 4i}[/tex]
Required
Solve
We have:
[tex]\frac{5i}{3 - 4i}[/tex]
Rationalize
[tex]\frac{5i}{3 - 4i} = \frac{5i}{3 - 4i} * \frac{3 + 4i}{3 + 4i}[/tex]
[tex]\frac{5i}{3 - 4i} = \frac{5i(3 + 4i)}{(3 - 4i)(3 + 4i)}[/tex]
Apply difference of two squares on the denominator
[tex]\frac{5i}{3 - 4i} = \frac{5i(3 + 4i)}{3^2 - (4i)^2}[/tex]
[tex]\frac{5i}{3 - 4i} = \frac{5i(3 + 4i)}{9 - (16*-1)}[/tex]
[tex]\frac{5i}{3 - 4i} = \frac{5i(3 + 4i)}{9 +16}[/tex]
[tex]\frac{5i}{3 - 4i} = \frac{5i(3 + 4i)}{25}[/tex]
Divide common factor (5)
[tex]\frac{5i}{3 - 4i} = \frac{i(3 + 4i)}{5}[/tex]
Expand the numerator
[tex]\frac{5i}{3 - 4i} = \frac{3i + 4*-1}{5}[/tex]
[tex]\frac{5i}{3 - 4i} = \frac{3i - 4}{5}[/tex]
