Question

The real part of $ {\left( {i - \sqrt 3 } \right)^{13}} $ is :
a) $ {2^{ - 10}}\sqrt 3 $
b) $ - {2^{12}}\sqrt 3 $
c) $ {2^{ - 12}}\sqrt 3 $
d) $ - {2^{ - 12}}\sqrt 3 $
e) $ - {2^{10}}\sqrt 3 $

Answer 1

Hint: We will use the cube root of unity to solve the problem. The given expression looks similar to one of the cube roots of unity ( $ {w^2} $ ) so we will reduce it to that with required algebraic manipulations. After that we will use the properties of the root to find the answer.
Formula used:
The solutions to the equation $ {x^3} - 1 = 0 $ are called the cube root of unity.
The cube roots of unity are $ 1,w,{w^2} $ where, $ w = \dfrac{{ - 1 + i\sqrt 3 }}{2},{w^2} = \dfrac{{ - 1 - i\sqrt 3 }}{2} $
1) $ {w^n} = {w^k} $ , where k is the remainder after dividing n with 3.
2) $ {i^n} = {i^k} $ , where k is the remainder after dividing n with 4 and $ i = \sqrt { - 1} $ .

Complete step-by-step answer:
The given expression is $ {\left( {i - \sqrt 3 } \right)^{13}} $
Let $ y = i - \sqrt 3 $
Now we will reduce this expression to $ {w^2} $ which is the cube root of unity.
Let’s multiply ‘I’ and divide ‘2’ in the above equation.
\[ \Rightarrow iy = - 1 - i\sqrt 3 \] [Multiplying ‘i’ throughout the equation]
\[ \Rightarrow \dfrac{{iy}}{2} = \dfrac{{ - 1 - i\sqrt 3 }}{2}\] [Dividinging 2 throughout the equation]
\[ \Rightarrow \dfrac{{iy}}{2} = {w^2}\] where $ {w^2} $ is one of the cube root of unity.
Now we have,
\[\dfrac{{iy}}{2} = {w^2}\]
\[ \Rightarrow iy = 2{w^2}\]
\[ \Rightarrow {\left( {iy} \right)^{13}} = {\left( {2{w^2}} \right)^{13}}\]
\[ \Rightarrow {i^{13}} \times {y^{13}} = {2^{13}}{\left( {{w^2}} \right)^{13}}\]
\[ \Rightarrow {i^{12}} \times i \times {y^{13}} = {2^{13}} \times {w^{16}}\]
\[ \Rightarrow 1 \times i \times {y^{13}} = {2^{13}} \times {w^2}\]
\[ \Rightarrow {y^{13}} = - i\left( {{2^{13}} \times {w^2}} \right)\] [Multiplying ‘-i’ throughout the equation]
As \[{w^2} = \dfrac{{ - 1 - i\sqrt 3 }}{2}\] so we can write
\[{y^{13}} = - i\left( {{2^{13}} \times \dfrac{{ - 1 - i\sqrt 3 }}{2}} \right)\]
\[ \Rightarrow {y^{13}} = - i\left[ {{2^{12}} \times \left( { - 1 - i\sqrt 3 } \right)} \right]\]
\[ \Rightarrow {y^{13}} = {2^{12}} \times \left( { - \sqrt 3 + i} \right)\]
\[ \Rightarrow {y^{13}} = - {2^{12}}\sqrt 3 + {2^{12}}i\]
\[ \Rightarrow \operatorname{Re} \left( {{y^{13}}} \right) = - {2^{12}}\sqrt 3 \]
Thus the real [art of the given expression is \[ - {2^{12}}\sqrt 3 \].
As option ‘b’ corresponds to the answer we obtained. Therefore, it is the correct choice.
So, the correct answer is “Option B”.

Note: While finding out the expansion of a complex expression, always try to reduce the expression to an expression known to us whose power calculation will be easy. Try not to use the binomial expansion as it is lengthy and time consuming.