Question

How do you simplify \[\dfrac{8+10i}{8+6i}\] and write it in $a+bi$ form?

Answer 1

Hint: We need to have a fair knowledge of complex numbers and the different equations and formulae that involves it. The general form of a complex number is \[a+ib\] , where the first part is the real term and the second term is the complex part. Here \[i\] represents iota and is defined as,
\[\begin{align}
  & i=\sqrt{-1} \\
 & \Rightarrow {{i}^{2}}=-1 \\
\end{align}\]
The easiest way to solve the given problem is to do rationalization of the denominator and then to separate the real and imaginary terms.

Complete step by step solution:
Now we start off with the solution of the given problem as,
We first rationalize the denominator by multiplying the numerator and denominator by the conjugate complex number of the denominator. We now rewrite the given problem as,
\[\begin{align}
  & \dfrac{8+10i}{8+6i} \\
 & \Rightarrow \dfrac{\left( 8+10i \right)\left( 8-6i \right)}{\left( 8+6i \right)\left( 8-6i \right)} \\
\end{align}\]
We now do multiplication of the complex numbers to get,
\[\Rightarrow \dfrac{64-48i+80i-60{{i}^{2}}}{64-36{{i}^{2}}}\]
Now, using the relation \[{{i}^{2}}=-1\] we write,
\[\begin{align}
  & \Rightarrow \dfrac{64-48i+80i-60\left( -1 \right)}{64-36\left( -1 \right)} \\
 & \Rightarrow \dfrac{64-48i+80i+60}{64+36} \\
 & \Rightarrow \dfrac{124+32i}{100} \\
\end{align}\]
Now, we separate the real and imaginary parts of the formed equation, to get the perfect solution, hence we write,
\[\begin{align}
  & \dfrac{124}{100}+\dfrac{32}{100}i \\
 & \Rightarrow \dfrac{31}{25}+\dfrac{8}{25}i \\
\end{align}\]

Thus, the answer to our problem is \[\dfrac{31}{25}+\dfrac{8}{25}i\] .

Note: For these types of problems, we need to remember and keep in mind the general form of complex numbers. The given problem is first solved by a simple rationalization of the denominator, followed by simple multiplication of two complex numbers and replacing the relation \[{{i}^{2}}=-1\] . After all these things we separate the real part and imaginary part and write the hence formed answer.