Question

If $(x+iy{)}^{1/3}=a+ib,$, then $\frac{x}{a}+\frac{y}{b}=$

• A. $4(a^2-b^2)$
• B. $0$
• C. $1$
• D. $noneofthese$

Answer 1

The correct option is A $4(a^2-b^2)$

${(x+iy)}^1/3=a+ib$

$\Rightarrow x+iy={(a+ib)}^3$

$=a^3-i{b}^3+i.3a^{2}b-3a{b}^2$

$=a^3-3a{b}^2+i(3a^{2}b-b^3)$

$\Rightarrow x=a^3-3a{b}^2;y=3a^{2}b-b^3$

$\Rightarrow \frac{x}{a}=a^2-3b^2;\frac{y}{b}=3a^2-b^2$

$\therefore \frac{x}{a}+\frac{y}{b}=4(a^2-b^2)$

Hence, the answer is $4(a^2-b^2).$