Answer :
Answer:
• (a)-i
,• (b)17+11i
Explanation:
Part A
We are to express the complex number i²³ in the simplest form.
[tex]\begin{gathered} First\text{ write the index as a sum of an even and an odd number} \\ i^{\mleft\{23\mright\}}=i^{22+1} \\ \text{Next, separate using the addition law of indices} \\ =i^{22}\times i^1 \end{gathered}[/tex]Then rewrite in the form below:
[tex]\begin{gathered} =(i^2)^{11}\times i^{} \\ U\sin g\text{ the fact that: }i^2=-1 \\ (i^2)^{11}\times i^{}=(-1)^{11}\times i=-1\times i=-i \end{gathered}[/tex]Therefore:
[tex]i^{23}=-i[/tex]Part B
Given the complex expression:
[tex]4\mleft(3+4i\mright)-5i\mleft(1+i\mright)[/tex]First, open the brackets:
[tex]\begin{gathered} =12+16i-5i-5i^2 \\ =12+11i-5i^2 \\ i^2=-1 \\ \implies=12+11i-5i^2=12+11i-5(-1) \\ =12+11i+5 \\ =12+5+11i \\ =17+11i \end{gathered}[/tex]The complex number in standard form is 17+11i.