Question

evaluate: [i^18+(1/i)^25]^3​

Answer 1

Step-by-step explanation:

$\{ {i}^{18} + ( \frac{1}{i} )^{25 } \} ^{3} \\ = \{ {(i^{2}) }^{9} + \frac{1}{(i)^{25 }} \} ^{3} \\ = \{ {( - 1) }^{9} + \frac{1}{(i)^{24 } \times i} \} ^{3} \\ = \{ - 1 + \frac{1}{(i^{2})^{12} \times i} \} ^{3} \\ = \{ - 1 + \frac{1}{( - 1)^{12} \times i} \} ^{3} \\= \{ - 1 + \frac{1}{1 \times i} \} ^{3} \\ = \{ - 1 + \frac{1}{ i} \} ^{3} \\= \{\frac{ - i + 1}{ i} \} ^{3} \\ = \frac{(1 - i)^{3} }{ {i}^{3} } \\ = \frac{1 - {i}^{3} - 3 {(1)}^{2} \times i + 3 \times1 \times {i}^{2} } { {i}^{2} \times i } \\ = \frac{1 - {i}^{2} \times i - 3i + 3( - 1) }{ - 1 \times i} \\ = \frac{1 - ( - 1)i - 3i - 3}{ - i} \\ = \frac{1 + i - 3i - 3}{ - i} \\ = \frac{ - 2 - 2i}{ - i} \\ = \frac{ - (2 + 2i)}{ - i} \\ = \frac{2 + 2i}{i} \\ = \frac{2}{i} + \frac{2i}{i} \\ = \frac{2}{i} + 2 \\ = 2 + 2 {i}^{ - 1}$