root(5+12i) + root(5-12i)/ root(5+12i) - root(5-12i) =

1. -3/2 i 2. 3/2i

3. -3/2 4. 3/2

the given expression is: 5+12i+5-12i5+12i-5-12i
to rationalise the denominator, multiply the numerator and denominator by 5+12i+5-12i .
5+12i+5-12i5+12i-5-12i.5+12i+5-12i5+12i+5-12i=5+12i+5-12i2(5+12i)-(5-12i)=5+12i+5-12i+2.5+12i.5-12i5+12i-5+12i=10+2(5+12i)(5-12i)24i=10+225-144i224i=10+225+14424i
=10+2.1324i    [25+144=169=13=10+2624i=3624i=32i

thus option (2) is correct.

hope this helps you
 

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