If $(x + i y)^{5} = p + i q$ , then prove that $(y + i x)^{5} = q + i p$ .

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Solution

$(x + i y)^{5} = p + i q$
or $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (x + i y)^{5} = ¯ ¯¯¯¯¯¯¯¯¯¯¯¯ ¯ p + i q$
or $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (x + i y)^{5} = p - i q$
or $(x - i y)^{5} = p - i q$
or $i^{5} (x - i y)^{5} = p i^{5} - i^{6} q$
or $(x i - i^{2} y)^{5} = p i + q$
or $(y + i x)^{5} = p i + q$
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