Find the quotient given below:
[tex]\frac{4+4i}{5+4i}[/tex]
When managing complex numbers, we must recall:
[tex]\begin{gathered} i^2=-1 \\ \text{ Or, equivalently:} \\ i=\sqrt{-1} \end{gathered}[/tex]
Multiply and divide the expression by the conjugate of the denominator:
[tex]\frac{4+4i}{5+4i}\cdot\frac{5-4i}{5-4i}[/tex]
Multiply the expressions in the numerator and in the denominator. We can apply the special product formula in the denominator:
[tex](a+b)(a-b)=a^2-b^2[/tex]
Operating:
[tex]\frac{(4+4i)(5-4i)}{5^2-(4i)^2}[/tex]
Operate and simplify:
[tex]\frac{20-16i+20i-16i^2}{25-16i^2}[/tex]
Applying the property mentioned above:
[tex]\frac{20-16i+20i+16}{25+16}[/tex]
Simplifying:
[tex]\frac{36+4i}{41}[/tex]
