Express the given complex number in the form | vedclass

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Question

(A) We use the identity $(x+y)^{3} = x^{3} + y^{3} + 3xy(x+y)$.Given expression: $\left(-2-\frac{1}{3}i\right)^{3} = -\left(2+\frac{1}{3}i\right)^{3}$

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) We use the identity $(x+y)^{3} = x^{3} + y^{3} + 3xy(x+y)$.Given expression: $\left(-2-\frac{1}{3}i\right)^{3} = -\left(2+\frac{1}{3}i\right)^{3}$$= -\left[2^{3} + \left(\frac{1}{3}i\right)^{3} + 3(2)\left(\frac{1}{3}i\right)\left(2+\frac{1}{3}i\right)\right]$$= -\left[8 - \frac{1}{27}i + 2i\left(2+\frac{1}{3}i\right)\right]$$= -\left[8 - \frac{1}{27}i + 4i + \frac{2}{3}i^{2}\right]$Since $i^{2} = -1$,we have:$= -\left[8 - \frac{1}{27}i + 4i - \frac{2}{3}\right]$$= -\left[\left(8-\frac{2}{3}\right) + \left(4-\frac{1}{27}\right)i\right]$$= -\left[\frac{22}{3} + \frac{107}{27}i\right]$$= -\frac{22}{3} - \frac{107}{27}i$

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

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