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Evaluate the Following:\[( I^{77} + I^{70} + I^{87} + I^{414} )^3\] - Mathematics

Evaluate the following:

\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]

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Solution

\[\left( i^{77} + i^{70} + i^{87} + i^{414} \right)^3 = \left( i^{4 \times 19 + 1} + i^{4 \times 17 + 2} + i^{4 \times 21 + 3} + i^{4 \times 103 + 2} \right)^3 \]

\[ = \left[ \left\{ \left( i^4 \right)^{19} \times i \right\} + \left\{ \left( i^4 \right)^{17} \times i^2 \right\} + \left\{ \left( i^4 \right)^{21} \times i^3 \right\} + \left\{ \left( i^4 \right)^{103} \times i^2 \right\} \right]\]

\[ = \left( i - 1 - i - 1 \right)^3 \left( \because i^4 = 1, i^3 = - i and i^2 = - 1 \right)\]

\[ = \left( - 2 \right)^3 \]

\[ = - 8\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.1 | Q 1.6 | Page 3
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