Question

Express in the form a + ib. (i) ${(5-3i)}^3$ (ii) $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}$

Answer 1

(i)

$(5-3i{)}^3=125+27i^3+225i+135i^2$

$(5-3i{)}^3=-10+198i$

(ii)

$\frac{5+\sqrt{2}i}{1-\sqrt{2}i}=\frac{5+\sqrt{2}i}{1-\sqrt{2}i}\times \frac{1+\sqrt{2}i}{1+\sqrt{2}i}$

$=\frac{5+5\sqrt{2}i+\sqrt{2}i+2i^2}{1^2-2i^2}$

$=\frac{5+6\sqrt{2}i-2}{1+2}$

$=\frac{3+6\sqrt{2}i}{3}$

$\frac{5+\sqrt{2}i}{1-\sqrt{2}i}=1+2\sqrt{2}i$