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$\\dfrac{1}{i} = \\dfrac{i}{i^2} = \\dfrac{i}{-1} = -i \\Rightarrow \\left(\\dfrac{1}{i}\\right)^{25} = (-i)^{25} = (-1)^{25} \\cdot i^{25} = -1 \\cdot i^{24} \\cdot i = -i$

$\\left[i^{18} + \\left(\\dfrac{1}{i}\\right)^{25}\\right]^3 = [-1 + (-i)]^3 = [-(1+i)]^3 = -(1+i)^3$

$(1+i)^2 = 1 + 2i + i^2 = 2i$

$(1+i)^3 = 2i(1+i) = 2i + 2i^2 = 2i - 2 = -2 + 2i$

$-(1+i)^3 = -(-2+2i) = 2 - 2i = 2(1-i)$

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Evaluate [i18+(i1​)25]3.

Solution

Relevant Questions

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