If (x+iy)2=7+24i, then the value of (7+√−576)12−(7−√−576)12 is:
Let (x+iy)2=7+24i ......(i)
⇒x2−y2+i(2xy)=7+24i
⇒x2−y2=7,xy=12
⇒(12y)2−y2=7
⇒144−y4=7y2
⇒y4+7y2−144=0
⇒y2=−7±252=−16 or 9
⇒y=±3
Taking conjugate of equation (i), we get
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(x+iy)2=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯7+24i
⇒(x−iy)2=7−24i ...... [∵¯¯¯¯¯z2=¯¯¯z2]
⇒(x−iy)=√7−24i
Now, (7+√−576)12−(7−√−576)12
=√7+24i−√7−24i
=x+iy−(x−iy)
=2yi=6i or −6i
Hence, option D is correct.