Question

Express the given equation in the form of a + ib, a, b ∈ R $i=\sqrt{-1}$. State the values of a and b.
$\frac{4i^{8}-3i^{9}+3}{3i^{11}-4i^{10}-2}$

Answer 1

As we know that

$\sqrt{ - 1} = i \\ \\ {i}^{2} = - 1 \\ \\ {i}^{4} = 1 \\ \\ {i}^{3} = - i$
$\frac{4i^{8}-3i^{9}+3}{3i^{11}-4i^{10}-2} \\ \\ \frac{4( { {i}^{4} )}^{2} -3i^{(2 \times 4 + 1)}+3}{3i^{4 \times 2 + 3}-4i^{4 \times 2 + 2}-2} \\ \\ = \frac{4 - 3i + 3}{3 {i}^{3} - 4 {i}^{2} - 2} \\ \\ = \frac{7 - 3i}{ - 3i + 4 - 2} \\ \\ = \frac{7 - 3i}{ 2 - 3i} \\ \\ rationalise \: the \: denominator \\ \\ = \frac{7 - 3i}{ 2 - 3i} \times \frac{(2 + 3i)}{(2 + 3i)} \\ \\ = \frac{14 + 21i - 6i + 9}{4 + 9} \\ \\ = \frac{23 + 15i}{13} \\ \\ a + ib = \frac{23}{13} + i \frac{15}{13} \\ \\ a = \frac{23}{13} \\ \\ b = \frac{15}{13}$
Hope it helps you.