Question

Find the modulus and amplitude of \[{{\left( 1+i\sqrt{3} \right)}^{8}}\] .
A. $256\text{, }\dfrac{\pi }{3}$
B. $256\text{, }\dfrac{2\pi }{3}$
C. $2\text{, }\dfrac{2\pi }{3}$
D. $256\text{, }\dfrac{8\pi }{3}$

Answer 1

Hint: First, we will start by defining what a complex number is and then we will learn to write the complex number in a polar form that is: $z=r\cos \theta +i\left( r\sin \theta \right)$. We will then take the complex number given in the question and multiply and divide it by 2 after that we will expand it and visualize it such that we get the polar form and finally we will compare it with the standard form and get the answer.

Complete step by step answer:
We are given \[{{\left( 1+i\sqrt{3} \right)}^{8}}\] which is a complex number. So, let’s first define what a complex number is.
So basically, complex numbers are the numbers that are expressed in the form of $a+ib$ where $'i'$ is an imaginary number called iota and has the value of $\sqrt{-1}$ . For example, $\left( 3+5i \right)$ is a complex number, where $3$ is a real number and $5i$ is an imaginary number. Therefore, the combination of both the real number and imaginary number is a complex number. Now, any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers and the numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result for example: \[\sqrt{-2},\sqrt{-7}\] etc. are all imaginary numbers.
Now, let $z=a+ib$ be a complex number. Now, the modulus of $z$ is represented as \[\left| z \right|\]. So, the value of \[\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\] .
Now, we all know that the pair of numbers $\left( x,y \right)$ can be represented on the XY-plane, where $x$ is called abscissa and $y$ is called the ordinate. Similarly, we can represent complex numbers also on a plane called complex plane. Similar to the X-axis and Y-axis in two-dimensional geometry, there are two axes in a complex plane.
The axis which is horizontal is called the real axis. The axis which is vertical is called the imaginary axis.
The complex number $z=a+ib$ which corresponds to the ordered pair $\left( a,b \right)$ is represented geometrically as the unique point $\left( a,b \right)$in the XY-plane.
Now, $z=a+ib$ can be written as $z=r\cos \theta +i\left( r\sin \theta \right)$ which is called the polar form of complex number.
Here, $r=\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$ is modulus of $z$ and $\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$ is known as the argument or amplitude of $z$ .

Now, we are given in the question to find the modulus and amplitude of \[{{\left( 1+i\sqrt{3} \right)}^{8}}\] , let’s first divide and multiply the given number by $2$ :
We have:
 \[\begin{align}
  & \Rightarrow z={{\left( 1+i\sqrt{3} \right)}^{8}} \\
 & \Rightarrow z={{\left( 2\times \left( \dfrac{1+i\sqrt{3}}{2} \right) \right)}^{8}}\Rightarrow z={{\left( 2\times \left( \dfrac{1}{2}+i\dfrac{\sqrt{3}}{2} \right) \right)}^{8}} \\
 & \Rightarrow z={{2}^{8}}{{\left( \dfrac{1}{2}+i\dfrac{\sqrt{3}}{2} \right)}^{8}} \\
\end{align}\]
Now, we know that $\cos \left( \dfrac{\pi }{3} \right)=\dfrac{1}{2}$ and $\sin \left( \dfrac{\pi }{3} \right)=\dfrac{\sqrt{3}}{2}$ , we will put this in above equation and hence we will get:
$z={{2}^{8}}{{\left( \cos \left( \dfrac{\pi }{3} \right)+i\sin \left( \dfrac{\pi }{3} \right) \right)}^{8}}$ , now we know that ${{e}^{i\theta }}=\cos \theta +i\sin \theta $ , therefore:
\[\begin{align}
  & \Rightarrow z={{2}^{8}}{{\left( \cos \left( \dfrac{\pi }{3} \right)+i\sin \left( \dfrac{\pi }{3} \right) \right)}^{8}}={{2}^{8}}{{\left( {{e}^{i\left( \dfrac{\pi }{3} \right)}} \right)}^{8}} \\
 & \Rightarrow z={{2}^{8}}{{e}^{\dfrac{8\pi }{3}i}}\Rightarrow z={{2}^{8}}{{e}^{\left( 2\pi +\dfrac{2\pi }{3}i \right)}} \\
 & \Rightarrow z={{2}^{8}}{{e}^{\left( \dfrac{2\pi }{3}i \right)}} \\
\end{align}\]
Now, we will again expand ${{e}^{\left( \dfrac{2\pi }{3}i \right)}}\Rightarrow {{e}^{\left( \dfrac{2\pi }{3}i \right)}}=\cos \dfrac{2\pi }{3}+i\sin \dfrac{2\pi }{3}$ , therefore:
$z={{2}^{8}}\left( \cos \dfrac{2\pi }{3}+i\sin \dfrac{2\pi }{3} \right)$ , on comparing it with the standard equation: $z=r\cos \theta +i\left( r\sin \theta \right)$ where $r=\left| z \right|$ ,
Therefore the modulus of \[{{\left( 1+i\sqrt{3} \right)}^{8}}\] is \[{{2}^{8}}=256\] and amplitude is $\dfrac{2\pi }{3}$.
Hence, option B is correct.

Note:
We can also find modulus by directly applying the formula here that is \[\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\], we will get by this: \[z={{\left( 1+i\sqrt{3} \right)}^{8}}\Rightarrow \left| z \right|=\sqrt{\left( {{1}^{2}} \right)+{{\left( \sqrt{3} \right)}^{2}}}=2\], we will raise it to the power 8 as initially we are given with this power therefore the modulus will be ${{2}^{8}}$ .