ok for a question like this what u would do first is try to establish that infact what they say is true, ie prove that the LHS = RHS. but note that if that was the supplied answer in the exam it would not be accepted, and u would lose marks as u did not do as the questioned asked. "prove", please learn all of the words that are involved in math questions ase they indicate what sort of direction you should take when answering the question. "prove", "deduce", "show" are all different depending on the context. for example, for "show" you are normally allowed to assume the result where as "prove" u are not. y am i doing this, cause it is helpful for such question. where two answers that use the same working and essentially the same difficulty and thinking process could be awarded different marks.
ok so for the actual answer;
my recommended answer:
LHS:
=sqrt(a+bi) + sqrt(a-bi)
=sqrt[ {sqrt(a+bi) + sqrt(a-bi)}^2 ]
=sqrt[ a+bi + 2sqrt{(a+bi)(a-bi)} +a-bi ]
=sqrt[ 2 { a + sqrt(a^2+b^2) } ]
=RHS
q.e.d
the more obvious but debatable answer: (do it this way at your own risk)
sqrt(a+bi) + sqrt(a-bi)=sqrt[ 2 { a + sqrt(a^2+b^2) } ]
square both sides:
a+bi + 2sqrt{(a+bi)(a-bi)} +a-bi = 2 { a + sqrt(a^2+b^2) }
therefore; 2 { a + sqrt(a^2+b^2) } = 2 { a + sqrt(a^2+b^2) }
therefore it is true. q.e.d
in no way do i enforce any method when answering a question, which ever way your comfortable with, and think will get you the full marks.
