# root(5+12i) + root(5-12i)/ root(5+12i) - root(5-12i) = 1. -3/2 i 2. 3/2i3. -3/2 4. 3/2

the given expression is: $\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$
to rationalise the denominator, multiply the numerator and denominator by $\sqrt{5+12i}+\sqrt{5-12i}$ .
$\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}.\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}+\sqrt{5-12i}}\phantom{\rule{0ex}{0ex}}=\frac{{\left[\sqrt{5+12i}+\sqrt{5-12i}\right]}^{2}}{\left(5+12i\right)-\left(5-12i\right)}\phantom{\rule{0ex}{0ex}}=\frac{5+12i+5-12i+2.\sqrt{5+12i}.\sqrt{5-12i}}{5+12i-5+12i}\phantom{\rule{0ex}{0ex}}=\frac{10+2\sqrt{\left(5+12i\right)\left(5-12i\right)}}{24i}\phantom{\rule{0ex}{0ex}}=\frac{10+2\sqrt{25-144{i}^{2}}}{24i}\phantom{\rule{0ex}{0ex}}=\frac{10+2\sqrt{25+144}}{24i}$

thus option (2) is correct.

hope this helps you

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