# Express the following in the standard form $a + ib$: $\large\frac{i^{4}+i^{9}+i^{16}}{3-2i^{8}-i^{10}-i^{15}}$

This is the fourth part of the multipart question Q1.

## 1 Answer

Toolbox:
• $i^2=-1,i^3=-i,i^4=1$.
• In general,$i^{4n-3}=i$,$i^{4n-2}=-1$,$i^{4n}=1$
• If $z=a+ib$ then ,
• $\bar{z}=a-ib$
• $\mid z\mid=\sqrt{a^2+b^2}$
• $z^{-1}=\large\frac{a-ib}{a^2+b^2}$
• Also $Re(z)=a,Im(z)=b$
$\large\frac{i^4+i^9+i^{16}}{3-2i^8-i^{10}-i^{15}}=\large\frac{1+i^8.i+1}{3-2-i^8.i^2-i^{12}.i^3}$
$\qquad\qquad\qquad=\large\frac{1+i+1}{1+1+i}$
$\qquad\qquad\qquad=\large\frac{2+i}{2+i}$
$\qquad\qquad\qquad=1$
$\qquad\qquad\qquad=1+0i$
answered Jun 7, 2013

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