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Standard XII

Mathematics

Combination

Evaluate the ...

Question

Evaluate the following
${(i^{77} + i^{70} + i^{87} + i^{414})}^{3}$

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Solution

$(i^{77} + i^{70} + i^{87} + i^{414})^{3}$

$∵ i^{4} = 1$

$[(i^{4})^{19} i + (i^{4})^{17} i^{2} + (i^{4})^{2} i^{3} + (i^{4})^{103} i^{2}]^{3}$

$\Rightarrow [i + i^{2} + i^{3} + i^{2}]^{3}$

$i^{2} = - 1$ and $i^{3} = - i$

$\Rightarrow [i + i^{3} - 1 - 1]^{3} = [i - i - 1 - 1]^{3}$

$\Rightarrow (- 2)^{3} = - 8$ .

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Evaluate the following(i77+i70+i87+i414)3

(i77+i70+i87+i414)3∵i4=1[(i4)19i+(i4)17i2+(i4)2i3+(i4)103i2]3⇒[i+i2+i3+i2]3i2=−1 and i3=−i⇒[i+i3−1−1]3=[i−i−1−1]3⇒(−2)3=−8.

Evaluate the following
${(i^{77} + i^{70} + i^{87} + i^{414})}^{3}$