Question

Evaluate the following:
\[{{\left( {{i}^{77}}+{{i}^{70}}+{{i}^{87}}+{{i}^{414}} \right)}^{3}}\]

Answer 1

Hint: Before solving this question, we must know about i or iota that it 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 can use other rules that are as follows
If the power of I or iota when divided by 4 and we get 0 as remainder, then we can say that the value would be 1.
If the power of I or iota when divided by 4 and we get 1 as remainder, then we can say that the value would be i.
If the power of I or iota when divided by 4 and we get 2 as remainder, then we can say that the value would be -1.
If the power of I or iota when divided by 4 and we get 3 as remainder, then we can say that the value would be -i.

As mentioned in the question, we have to evaluate the given expression that is given to us. Now, on using the information that is provided in the hint, we can write the following as
\[\begin{align}
  & ={{\left( {{i}^{77}}+{{i}^{70}}+{{i}^{87}}+{{i}^{414}} \right)}^{3}} \\
 & ={{\left( {{i}^{4\times 19+1}}+{{i}^{4\times 17+2}}+{{i}^{4\times 21+3}}+{{i}^{4\times 103+2}} \right)}^{3}} \\
 & ={{\left( {{i}^{1}}+{{i}^{2}}+{{i}^{3}}+{{i}^{2}} \right)}^{3}} \\
\end{align}\]
We know that,
$
i=\sqrt{-1} \\
{{i}^{2}}=-1 \\
{{i}^{3}}=-i \\
{{i}^{4}}=1 \\
$
After substituting, we get
$
= {{\left( i+\left( -1 \right)+-i+\left( -1 \right) \right)}^{3}} \\
 = {{(-2)}^{3}}=-8 \\
$
(Because
If the power of I or iota when divided by 4 and we get 0 as remainder, then we can say that the value would be 1.
If the power of I or iota when divided by 4 and we get 1 as remainder, then we can say that the value would be i.
If the power of I or iota when divided by 4 and we get 2 as remainder, then we can say that the value would be -1.
If the power of I or iota when divided by 4 and we get 3 as remainder, then we can say that the value would be -i.)
Hence, the correct answer is -8.

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.
Also, the information that is given in the hint related to the power having a factor as the mentioned numbers is also very important as without it one could never get to the correct answer.