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  • Express each of the complex number in the form a+ib. (10) (-2-1/3i)^3

Express each of the complex number in the form  a+ib.

Q : 10     \left ( -2-\frac{1}{3}i \right )^3

Answers (1)

best_answer

Given problem is 
\left ( -2-\frac{1}{3}i \right )^3
Now, we will reduce it into

\left ( -2-\frac{1}{3}i \right )^3=-\left ( (2)^3+\left ( \frac{1}{3}i \right )^3 +3.(2)^2\frac{1}{3}i+3.\left ( \frac{1}{3}i \right )^2.2 \right )                (using \ (a+b)^3=a^3+b^3+3a^2b+3ab^2)
                           =-\left ( 8+\frac{1}{27}i^3+3.4.\frac{1}{3}i+3.\frac{1}{9}i^2.2 \right )                                                        
                           =-\left ( 8+\frac{1}{27}(-i)+4i+\frac{2}{3}(-1) \right )                                                               (\because i^3=-i \ and \ i^2 = -1)
                           =-\left ( 8-\frac{1}{27}i+4i-\frac{2}{3} \right )
                           =-\left ( \frac{(-1+108)}{27}i+\frac{24-2}{3} \right )
                           =-\frac{22}{3}-i\frac{107}{27}
                                                                         
Therefore, the answer is  -\frac{22}{3}-i\frac{107}{27}

Posted by

Gautam harsolia

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