Question

If $ {\left( {x + iy} \right)^5} = p + iq $ , then prove that $ {\left( {y + ix} \right)^5} = q + ip. $

Answer 1

Hint: The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if $ a $ and $ b $ are real, then) the then the complex conjugate of $ \left( {a + bi} \right) $ is equal to $ \left( {a - bi} \right) $ .

Complete step-by-step answer:
Given equation in the question is,
 $ \Rightarrow $ $ {\left( {x + iy} \right)^5} = p + iq $
We can write it as,
 $
   \Rightarrow \dfrac{1}{{{{\left( {x + iy} \right)}^5}}} = \dfrac{1}{{(p + iq)}} \\
   \Rightarrow \dfrac{1}{{{{(x + iy)}^5}}} = p - iq \\
   \Rightarrow {\left( {x - iy} \right)^5} = p - iq \;
   $
Multiply by $ {i^5} $ on both the sides,
 $
   \Rightarrow {i^5}{(x - iy)^5} = p{i^5} - q{i^6} \\
   \Rightarrow {(x{i^{}} - {i^2}y)^5} = pi + q \\
   \Rightarrow {(y + ix)^5} = pi + q. \;
  $
Hence proved.

Note: $ \Rightarrow $ Each of two complex numbers having their real parts identical and their imaginary parts of equal magnitude but opposite sign.
 $ \Rightarrow $ Conjugate of the conjugate of a complex number $ Z $ is the complex number itself.
 $ \Rightarrow $ Conjugate of the sum of two complex numbers $ {z_1},{z_2} $ is the sum of their conjugates.
\[\overline {z_1 + z_2} = \bar z_1 + \bar z_2.\]