( 13 + 3i )^3 = ( 13)^3 + ( 3i )^3 + 3 ( 13) ( 3i ) ( 13 + 3i ) = 127 + 27 i^3 + 3i ( 1 + 9i3) = 127 27i + i + 9 i^2 = 127 26i 9 ...

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Complex Numbers

Simplify: 1...

Question

Simplify: ${(\frac{1}{3} + 3 i)}^{3}$

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Solution

${(\frac{1}{3} + 3 i)}^{3}$
$= {(\frac{1}{3})}^{3} + {(3 i)}^{3} + 3 (\frac{1}{3}) (3 i) (\frac{1}{3} + 3 i)$
$= \frac{1}{27} + 27 i^{3} + 3 i (\frac{1 + 9 i}{3})$
$= \frac{1}{27} - 27 i + i + 9 i^{2}$
$= \frac{1}{27} - 26 i - 9$
$= - (\frac{242}{27} + 26 i)$

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