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.
$\overline {z_1 + z_2} = \bar z_1 + \bar z_2.$