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Algebra of Complex Numbers

Express 2+3 i...

Question

Express $(2 + 3 i)^{3}$ in the form $(a + i b)$

- 46 + 9 i

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- 40 + 7 i

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46 - 9 i

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46 + 9 i

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Solution

The correct option is A $- 46 + 9 i$
We know that
$(z_{1} + z_{2})^{3} = z_{1}^{3} + z_{2}^{3} + 3 z_{1} z_{2} (z_{1} + z_{2}) .$
$∵ (2 + 3 i)^{3} = 2^{3} + (3 i)^{3} + 3 \times 2 \times 3 i (2 + 3 i)$
$= 8 + 27 i^{3} + 36 i + 54 i^{2}$
$= 8 - 27 i + 36 i - 54 [∵ i^{3} = - i, i^{2} = - 1]$
$= - 46 + 9 i$

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