Respuesta :
The process of multiplying complex numbers is very similar when we multiply two binomials.
The only difference is the introduction of the expression:
[tex]\sqrt[]{-1}\text{ = i}[/tex]For example,
Multiply 2i by 8i
[tex]\begin{gathered} 2i\text{ }\times\text{ 8i = 2}\times\text{ 8}\times\text{ i}\times\text{ i} \\ =\text{ 16}\times i^2 \\ =\text{ 16 }\times\text{ -1} \\ =\text{ -}16 \end{gathered}[/tex]We must remember the following when multiplying imaginary numbers:
[tex]\begin{gathered} \sqrt[]{-1}\text{ = i} \\ i^2\text{ = -1} \end{gathered}[/tex]Let's work on another example:
(2i)^4:
[tex]\begin{gathered} =(2i)^4 \\ =\text{ 2i }\times\text{ 2i }\times\text{ 2i }\times\text{ 2i} \\ =\text{ 2}\times2\times2\times2\text{ }\times i\times i\times i\times i \\ =\text{ 16 }\times i^2\text{ }\times i^2 \\ =\text{ 16 }\times\text{ -1}\times-1 \\ =\text{ 16} \end{gathered}[/tex]Remember that:
[tex]\begin{gathered} i^2\text{ = -1} \\ i^3=i^2\text{ }\times\text{ i} \\ =\text{ -i} \\ i^4=i^2\text{ }\times i^2 \\ =\text{ -1 }\times\text{ -1} \\ =\text{ 1} \end{gathered}[/tex]Anytime we raise i to any power, the result changes depending on the value of the power. We can obtain the result by evaluating the terms as shown above.
