Question:
Express each of the following in the form (a + ib):
$\frac{(-2+5 i)}{(3-5 i)}$
Solution:
Given: $\frac{-2+5 i}{3-5 i}$
Now, rationalizing
$=\frac{-2+5 i}{3-5 i} \times \frac{3+5 i}{3+5 i}$
$=\frac{(-2+5 i)(3+5 i)}{(3-5 i)(3+5 i)} \ldots(\mathrm{i})$
Now, we know that,
$(a+b)(a-b)=\left(a^{2}-b^{2}\right)$
So, eq. (i) become
$=\frac{(-2+5 i)(3+5 i)}{(3)^{2}-(5 i)^{2}}$
$=\frac{-2(3)+(-2)(5 i)+5 i(3)+5 i(5 i)}{9-25 i^{2}}$
$=\frac{-6-10 i+15 i+25 i^{2}}{9-25(-1)}\left[\because i^{2}=-1\right]$
$=\frac{-6+5 i+25(-1)}{9+25}$
$=\frac{-31+5 i}{34}$
$=-\frac{31}{34}+\frac{5}{34} i$
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