Question

Simplify the given and express in the form of a + ib.
$\frac{3i^{5}+2i^{7}+i^{9}}{i^{6}+2i^{8}+3i^{18}}$

Answer 1

To simplify the given expression,use properties of iota

${i}^{2} = - 1 \\ \\ {i}^{4} = 1 \\ \\ {i}^{3} = - i$
$= \frac{3i^{5}+2i^{7}+i^{9}}{i^{6}+2i^{8}+3i^{18}} \\ \\ = \frac{3i^{4}i+2i^{4} {i}^{3} +i^{4 \times 2}i}{i^{4} {i}^{2} +2i^{4 \times 2}+3i^{4 \times 4 + 2}} \\ \\ = \frac{3i - 2i + i}{ - 1 + 2 + 3 ( - 1)} \\ \\ = \frac{2i}{ - 2} = - i \\ \\ \frac{3i^{5}+2i^{7}+i^{9}}{i^{6}+2i^{8}+3i^{18}} = - i\\\\a+ib=0-i \\ \\a=0\\\\b=-1$
Hope it helps you.