Express the given complex number in the form | vedclass

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Question

(A) Using the identity $(x+y)^{3} = x^{3} + y^{3} + 3xy(x+y)$:$\left(\frac{1}{3}+3i\right)^{3} = \left(\frac{1}{3}\right)^{3} + (3i)^{3} + 3\left(\fra

Vedclass · Accepted Answer

$\frac{-242}{27}-26i$

Vedclass · Answer

(A) Using the identity $(x+y)^{3} = x^{3} + y^{3} + 3xy(x+y)$:$\left(\frac{1}{3}+3iight)^{3} = \left(\frac{1}{3}ight)^{3} + (3i)^{3} + 3\left(\frac{1}{3}ight)(3i)\left(\frac{1}{3}+3iight)$$= \frac{1}{27} + 27i^{3} + 3i\left(\frac{1}{3}+3iight)$$= \frac{1}{27} - 27i + i + 9i^{2} \quad [	ext{since } i^{3} = -i, i^{2} = -1]$$= \frac{1}{27} - 27i + i - 9$$= \left(\frac{1}{27} - 9ight) + i(-27 + 1)$$= \frac{1-243}{27} - 26i$$= \frac{-242}{27} - 26i$

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

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