Part A
The given expression is:
[tex](-5(1+2i)+3i(3-4i)[/tex]
We expand to get:
[tex]-5-10i+9i-12i^2[/tex]
Note that [tex]i^2=-1[/tex]
[tex]12-5-10i+9i[/tex]
[tex]-5-10i+9i+12[/tex]
[tex]7-i[/tex]
Part Bi)
The given expression is;
[tex]\sqrt{-50}[/tex]
We simplify to get:
[tex]\sqrt{25\times 2\times -1}[/tex]
[tex]\sqrt{25\times} \sqr{2}\times \sqrt{-1}[/tex]
Note that:
[tex]\sqrt{-1}=i[/tex]
[tex]5\sqr{2}i[/tex]
This is now of the form [tex]a+bi[/tex], where [tex]a=0,b=5\sqr{2}[/tex].
This explains why it is a complex number;
Part bii)
The given expression is :
[tex]\frac{\sqrt{-50} }{\sqrt{5}}[/tex]
We simplify to get
[tex]\sqrt{\frac{-50 }{5}}[/tex]
[tex]\sqrt{-25}[/tex]
[tex]\sqrt{-25}=\sqrt{25}\times \sqrt{-1}[/tex]
[tex]\sqrt{-25}=5\times i[/tex]
[tex]\sqrt{-25}=5i[/tex]
Part C
We want to simplify:
[tex]i^{22}[/tex]
We rewrite in terms of [tex]i^2[/tex]
[tex]i^{22}=(i^2)^{11}[/tex]
[tex]i^{22}=(-1)^{11}[/tex]
[tex]i^{22}=-11[/tex]
