Evaluate: [i^ 18 + (1 i )^ 25 ]^ 3 . - Vidyadip

Question

Evaluate: $\bigg[\text{i}^{18}+\Big(\frac{1}{\text{i}}\Big)^{25}\bigg]^{3}.$

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Answer

$\bigg[\text{i}^{18}+\Big(\frac{1}{\text{i}}\Big)^{25}\bigg]^{3}$ $=\Big[\text{i}^{4\times4+2}+\frac{1}{\text{i}^{4\times6+1}}\Big]^3$ $=\bigg[\big(\text{i}^4\big)^4.\text{i}^2+\frac{1}{(\text{i}^4)^6.\text{i}}\bigg]^3$ $=\Big[\text{i}^2+\frac{1}{\text{i}}\Big]^3\ \ [\text{i}^4=1]$ $=\Big[-1+\frac{1}{\text{i}}\times\frac{\text{i}}{\text{i}}\Big]^3\ \ [\text{i}^2=-1]$ $=\Big[-1+\frac{\text{i}}{\text{i}^2}\Big]^3$ $=\big[-1-\text{i}\big]^3$ $=\big(-1\big)^3\big[1+\text{i}\big]^3$ $=-\big[1^3+\text{i}^3+3.1.\text{i}\big(1+\text{i}\big)\big]$ $=-\big[1+\text{i}^3+3\text{i}+3\text{i}^2\big]$ $=-\big[1-\text{i}+3\text{i}-3\big]$ $=-\big[-2+2\text{i}\big]$$=2-2\text{i}$

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