Express the given complex number in the form a + ib:
Question:

Express the given complex number in the form a + ib$\left(-2-\frac{1}{3} i\right)^{3}$

Solution:

$\left(-2-\frac{1}{3} i\right)^{3}=(-1)^{3}\left(2+\frac{1}{3} i\right)^{3}$

$=-\left[2^{3}+\left(\frac{i}{3}\right)^{3}+3(2)\left(\frac{i}{3}\right)\left(2+\frac{i}{3}\right)\right]$

$=-\left[8+\frac{i^{3}}{27}+2 i\left(2+\frac{i}{3}\right)\right]$

$=-\left[8-\frac{i}{27}+4 i+\frac{2 i^{2}}{3}\right] \quad\left[i^{3}=-i\right]$

$=-\left[8-\frac{i}{27}+4 i-\frac{2}{3}\right] \quad\left[i^{2}=-1\right]$

$=-\left[\frac{22}{3}+\frac{107 i}{27}\right]$

$=-\frac{22}{3}-\frac{107}{27} i$

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