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NCERT Class XI Mathematics - Complex Numbers and Quadratic Equations - Solutions

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Question : 33 of 52
Marks: +1, -0
Evaluate: [i18+(1i)25]3\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}
Solution:  
We have , [i18+(1i)25]3\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3} = [(i2)9+1(i2)12⋅1i]3\left[(i^2)^9+\frac{1}{(i^2)^{12}}\cdot\frac{1}{i}\right]^{3}
= [(−1)9+1(−1)12⋅1i]3\left[(-1)^9+\frac{1}{(-1)^{12}}\cdot\frac{1}{i}\right]^{3} = [−1+1i]3\left[-1+\frac{1}{i}\right]^{3}
= (−1)3+(1i)3(-1)^3+\left(\frac{1}{i}\right)^3 + 3(−1)2(1i)3(-1)^2\left(\frac{1}{i}\right) + 3(−1)(1i)23(-1)\left(\frac{1}{i}\right)^2
= - 1 + 1−i+3i−3i2\frac{1}{-i}+\frac{3}{i}-\frac{3}{i^2} = - 1 + −1i+3i−3−1\frac{-1}{i}+\frac{3}{i}-\frac{3}{-1}
= - 1 + 3 + 1i\frac{1}{i} (3 - 1) = 2 + 2i×ii\frac{2}{i}\times\frac{i}{i}
= 2 - 2i
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