Question

Find the square root of 4i.

Answer 1

Hint: Before solving this question, we must know that I or iota is nothing but \[\sqrt{-1}\] .
Now, from this conversion, we can see that we can generalize a few relations that are related to the power of i or iota and those relations are as follows
\[\begin{align}
  & i=\sqrt{-1}=i \\
 & {{i}^{2}}=-1 \\
 & {{i}^{3}}=-i \\
 & {{i}^{4}}=1 \\
\end{align}\]

Complete step-by-step answer:
Now, for solving this particular question, we will first assign the root of the given number of which root is to be found, to the general form of a complex number that is a+ib.
Then, we will solve further to get to the correct solution by getting the values of a and b.
As mentioned in the question, we have to evaluate the square root of the given complex number. Now, on using the information that is provided in the hint, we can write the square root of the given number to be equal to the a+ib as follows
\[\sqrt{\left( 4i \right)}=a+ib\]
Now, on squaring both sides of the equation, we get the following
\[\begin{align}
  & \sqrt{\left( 4i \right)}=a+ib \\
 & {{\left( a+ib \right)}^{2}}=4i \\
 & \left[ {{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy \right] \\
 & {{a}^{2}}+{{b}^{2}}{{i}^{2}}+2abi=4i \\
 & {{a}^{2}}-{{b}^{2}}+2abi=4i \\
\end{align}\]
Now, on comparing the real part with the real and the imaginary part with the imaginary one, we get
\[\begin{align}
  & {{a}^{2}}={{b}^{2}}\ and\ 4=2ab \\
 & a=\pm b\ and\ 2=ab\ \ \ ...(b) \\
\end{align}\]
Now, we can see from equation (b) that a should be equal to b rather than –b because if they both have different signs then the outcome cannot be positive and which is the actual case
Therefore, we can write the following
\[\begin{align}
  & a\times a=2 \\
 & a=\pm \sqrt{2} \\
\end{align}\]
Hence, we get the value of the root of the number as \[\pm \sqrt{2}+i\left( \pm \sqrt{2} \right)\] .

NOTE: The students can make an error if they don’t know the relations that are mentioned in the hint because without knowing those relations one can never get to the right answer. It is also very important to know how to begin with the question which should be just like the way it is mentioned in the hint.
Also, the information that is given in the hint related to the power having a factor as the mentioned numbers is also very important.
Also not only in this question, the students must be very careful while solving any such questions as if there is any mistake in the calculus, then the answer can come out to be wrong.