Question

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

Answer 1

${\displaystyle \frac{4{\text{i}}^8-3{\text{i}}^9+3}{3{\text{i}}^{11}-4{\text{i}}^{10}-2}=\frac{4{\left({\text{i}}^4\right)}^2-3{\left({\text{i}}^4\right)}^2\cdot \text{i}+3}{3{\left({\text{i}}^4\right)}^2\cdot {\text{i}}^3-4{\left({\text{i}}^4\right)}^2\cdot {\text{i}}^2-2}}$

Since, i² = – 1, i³ = – i and i⁴ = 1

∴ ${\displaystyle \frac{4{\text{i}}^8-3{\text{i}}^9+3}{3{\text{i}}^{11}-4{\text{i}}^{10}-2}=\frac{4{\left(1\right)}^2-3{\left(1\right)}^2\cdot \text{i}+3}{3{\left(1\right)}^2(-\text{i})-4{\left(1\right)}^2(-1)-2}}$

= ${\displaystyle \frac{4-3\text{i}+3}{-3\text{i}+4-2}}$

= ${\displaystyle \frac{7-3\text{i}}{2-3\text{i}}}$

= ${\displaystyle \frac{(7-3\text{i})(2+3\text{i})}{(2-3\text{i})(2+3\text{i})}}$

= ${\displaystyle \frac{14+21\text{i}-6\text{i}-9{\text{i}}^2}{4-9(-1)}}$

= ${\displaystyle \frac{14+15\text{i}-9(-1)}{4-9(-1)}}$

= ${\displaystyle \frac{23+15\text{i}}{13}}$

∴ ${\displaystyle \frac{4{\text{i}}^8-3{\text{i}}^9+3}{3{\text{i}}^{11}-4{\text{i}}^{10}-2}=\frac{23}{13}+\frac{15}{13}\text{i}}$

∴ a = ${\displaystyle \frac{23}{13}\text{and}\text{b}=\frac{15}{13}}$