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Complex Numbers

Perform the i...

Question

Perform the indicated operations.
$(- 1 + \sqrt{3 i})^{3} .$

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Solution

$(a + b)^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
$(- 1 + \sqrt{3 i})^{3} = (- 1)^{3} + (\sqrt{3 i}) + (\sqrt{3 i}) + (\sqrt{3 i}) + 3 (- 1)^{2} (\sqrt{3 i}) + 3 (- 1) (\sqrt{3 i})^{2}$
$= - 1 + 3 i \sqrt{3 i} + 3 \sqrt{3 i} - 3 \times 3 i$
$= - 1 + 3 i \sqrt{3 i} + 3 \sqrt{3 i} - 9 i$

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