Use the Binomial Theorem to expand the following complex number. Write your answer in standard form.
$$\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$$
A) $\frac { 36 } { 343 } - \frac { 20 \sqrt { 7 } } { 343 } i$
B) $\frac { 36 } { 343 } + \frac { 20 \sqrt { 7 } } { 49 } i$
C) $- \frac { 36 } { 49 } - \frac { 20 \sqrt { 7 } } { 343 } i$
D) $- \frac { 27 } { 343 } + \frac { \sqrt { 7 } } { 49 } i$
E) $- \frac { 27 } { 343 } - \frac { \sqrt { 7 } } { 49 } i$

Question

Quizplus · Accepted Answer

To expand $\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$ using the Binomial Theorem, we calculate each term of the expansion. The real part comes from the sum of the terms that involve even powers of $i$ (since $i^2 = -1$, $i^4 = 1$, etc.), and the imaginary part comes from the terms that involve odd powers of $i$.The expansion is:$$\left( - \frac { 3 } { 7 } \right)^3 + 3\left( - \frac { 3 } { 7 } \right)^2\left( - \frac { \sqrt { 7 } } { 7 } i \right) + 3\left( - \frac { 3 } { 7 } \right)\left( - \frac { \sqrt { 7 } } { 7 } i \right)^2 + \left( - \frac { \sqrt { 7 } } { 7 } i \right)^3$$Calculating each term:1. $\left( - \frac { 3 } { 7 } \right)^3 = -\frac{27}{343}$2. $3\left( - \frac { 3 } { 7 } \right)^2\left( - \frac { \sqrt { 7 } } { 7 } i \right) = 3 \cdot \frac{9}{49} \cdot - \frac { \sqrt { 7 } } { 7 } i = -\frac{27\sqrt{7}}{343}i$3. $3\left( - \frac { 3 } { 7 } \right)\left( - \frac { \sqrt { 7 } } { 7 } i \right)^2 = 3 \cdot -\frac{3}{7} \cdot \frac{7}{49}(-1) = \frac{9}{49}$4. $\left( - \frac { \sqrt { 7 } } { 7 } i \right)^3 = -\frac{7\sqrt{7}}{343}i^3 = -\frac{7\sqrt{7}}{343}(-i) = \frac{7\sqrt{7}}{343}i$Adding these together:Real part: $-\frac{27}{343} + \frac{9}{49} = \frac{-27 + 63}{343} = \frac{36}{343}$Imaginary part: $-\frac{27\sqrt{7}}{343}i + \frac{7\sqrt{7}}{343}i = -\frac{20\sqrt{7}}{343}i$Thus, the expansion in standard form is $\frac { 36 } { 343 } - \frac { 20 \sqrt { 7 } } { 343 } i$.

Use the Binomial Theorem to Expand the Following Complex Number (−37−77i)3\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }(−73​−77​​i)3

Use the Binomial Theorem to Expand the Following Complex Number $\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$