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  4. If z =2+3 i, then z 5+( barz)5 is equal to :

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Q. If $z =2+3 i$, then $z ^5+(\bar{z})^5$ is equal to :

JEE MainJEE Main 2022Complex Numbers and Quadratic Equations

Solution:

$ z^5+(\bar{z})^5=(2+3 i)^5+(2-3 i)^5 $
$ =2\left({ }^5 C_0 2^5+{ }^5 C_2 2^3(3 i)^2+{ }^5 C_4 2^1(3 i)^4\right) $
$ =2(32+10 \times 8(-9)+5 \times 2 \times 81)=244$