Question

Simplify:
$\frac{\sqrt{\left(5+12i\right)}+\sqrt{\left(5-12i\right)}}{\sqrt{\left(5+12i\right)}-\sqrt{\left(5-12i\right)}}$

• A. −32i−32i
• B. $\frac{3}{i}$
• C. $\frac{5}{2i}$
• D. $\frac{3}{2i}$

Answer 1

Answer: (D)

$\frac{3}{2i}$

We have,

$\frac{\sqrt{(5+12i)}+\sqrt{(5-12i)}}{\sqrt{(5+12i)}-\sqrt{(5-12i)}}$

$\frac{\sqrt{(5+12i)}+\sqrt{(5-12i)}}{\sqrt{(5+12i)}-\sqrt{(5-12i)}}\times \frac{\sqrt{(5+12i)}+\sqrt{(5-12i)}}{\sqrt{(5+12i)}+\sqrt{(5-12i)}}onrationlized$

$=\frac{{(\sqrt{(5+12i)}+\sqrt{(5-12i)})}^2}{{\left(\sqrt{(5+12i)}\right)}^2-{\left(\sqrt{(5-12i)}\right)}^2}$

$=\frac{5+12i+5-12i+2\sqrt{(5+12i)}\sqrt{(5-12i)}}{5+12i-5+12i}$

$=\frac{10+2\sqrt{25-144\times (-1)}}{24i}$

$=\frac{2(5+\sqrt{25+144})}{24i}$

$=\frac{5+\sqrt{169}}{12i}$

$=\frac{5+13}{12i}$

$=\frac{18}{12i}$

$=\frac{3}{2i}$

Hence, this is the answer.