Question

Perform the indicated operation and express the result as a simplified complex number in the form a+bi. Do not put any spaces between your characters. If your answer includes a number that is not an integer type it as a decimal rounded to the nearest hundredth. \frac{2+3i}{2-3i}simplifies to a+bi where:a= Answerb= Answer

Answer 1

Given:

`(2+3i)/(2-3i)`

Simplify,

`\begin{gathered} (2+3i)/(2-3i) \\ \text{Apply complex arithmatic rule,} \\ (2+3i)/(2-3i)=(2+3i)/(2-3i)*(2+3i)/(2+3i) \\ =((2+3i)^2)/(2^2-(3i)^2) \\ =(2^2+2(6i)+(3i)^2)/(4-(-9)) \\ =(4+12i-9)/(13) \\ =(-5+12i)/(13) \\ =-(5)/(13)+(12)/(13)i \end{gathered}`

The simplified form as a+bi is,

`\begin{gathered} a=-(5)/(13) \\ b=(12)/(13) \end{gathered}`