Question

How do you find \[{{\left( 1+i \right)}^{4}}\] ?

Answer 1

Hint: In the above mentioned problem \[i\] is basically an imaginary number also known as “iorta”. It is defined as,
\[\begin{align}
  & i=\sqrt{-1} \\
 & \Rightarrow {{i}^{2}}=-1 \\
\end{align}\]
The simplest method of solving these type of problems are by simple algebraic factorization. In our case, we will first find the value of \[{{\left( 1+i \right)}^{2}}\] , and then square it again to obtain our final desired result. In the given problem \[1\] is the real part, whereas \[i\] is the imaginary part.

Complete step by step answer:
Now, starting off with our solution,
As mentioned above, we will first try to find the value of \[{{\left( 1+i \right)}^{2}}\] . Evaluating step by step and first squaring \[\left( 1+i \right)\] we get the following,
\[\begin{align}
  & {{\left( 1+i \right)}^{2}} \\
 & \Rightarrow 1+2i+{{i}^{2}} \\
\end{align}\]
Now as mentioned previously that, \[{{i}^{2}}=-1\] , we put this value in the above equation, and we get,
\[1+2i+\left( -1 \right)\]
Now adding the real and imaginary parts, we get,
\[\begin{align}
  & \left( 1-1 \right)+2i \\
 & \Rightarrow 0+2i \\
 & \Rightarrow 2i \\
\end{align}\]
Now, according to the requirement of the problem, we need to square the result further, in order to obtain the desired result. The intermediate answer which we have obtained contains no real part and only contains the imaginary part which is equal to \[2i\] . Now squaring \[2i\] , we get our required answer, we square and get,
\[\begin{align}
  & {{\left( 2i \right)}^{2}} \\
 & \Rightarrow {{2}^{2}}{{i}^{2}} \\
 & \Rightarrow 4{{i}^{2}} \\
\end{align}\]
Now, as we have already told that \[{{i}^{2}}=-1\] , we directly put this known result in our equation, which then results to,
\[\begin{align}
 & \Rightarrow 4\left( -1 \right) \\
 & \Rightarrow -4 \\
\end{align}\]

Thus our answer to this problem is \[-4\].

Note: While solving these types of problems, firstly we have to keep in mind of the values of the real and imaginary part. We should not mix up both the parts and be careful while adding and subtracting. The given problem can also be done by using the binomial expansion formula.