Question

Simplify the expression √¯-25 over (5 - 2i) + (1 - 4i)

Answer 1

\[\frac{\sqrt{-25}}{(5 - 2i) + (1 - 4i)}\]
Consider the denominator,
(5-2i) + (1-4i) = 6 - 6i

Consider the numerator, \(\sqrt{-25}=5i\)

So,
\[\frac{\sqrt{-25}}{(5 - 2i) + (1 - 4i)}\]\[=\frac{5i}{6 - 6i}\]\[=\frac{5i}{6(1 - i)}\]Multiply the fraction by a conjugate \(\frac{1+i}{1+i}\)
\[=\frac{5i}{6(1 - i)}\times \frac{1+i}{1+i}\]\[=\frac{5i(1+i)}{6(1 - i)(1+i)}\]\[=\frac{5i+5i^2}{6(1^2 - (-1))}\]\[=\frac{5i+5(-1)}{6(2)}\]\[=\frac{5i-5}{12}\]