Question

What is the value of \[{{i}^{34}}\]?

Answer 1

Hint: For solving this question you should know about the value of \[{{i}^{2}}\]. If we want to solve this then we can make the factor form of this and we can also write it as a power of \[{{i}^{2}}\]. And if the power is multiplied with that then the answer will be determined for this.

Complete step-by-step answer:
As our question asked us to determine the value of \[{{i}^{34}}\].
As we know that the value of \[{{i}^{2}}=-1\] which is negative value and we will write our term \[{{i}^{34}}\] as a form of power of \[{{i}^{2}}\].
So, for this we will do factors of \[{{i}^{34}}\] and then we will solve this.
Now, if we assume that \[{{i}^{2}}=t\] then we can write \[{{i}^{34}}={{\left( {{i}^{2}} \right)}^{17}}\] and \[{{i}^{2}}=t\].
So, it can be written as: \[{{i}^{34}}={{t}^{17}}\] and we know that \[{{i}^{2}}=-1\].
So, the value of \[t=-1\].
We can write it as: \[{{i}^{34}}={{\left( -1 \right)}^{17}}\]
As we know that the even power of -1 gives and 1 and odd power of -1 gives us the value -1.
So, here the power is 17 which is an odd number.
So, the value of \[{{i}^{34}}=-1\].

Note: Alternate method:
We can check this value or we can determine the value of \[{{i}^{34}}\] by this method also:
We can write \[{{i}^{34}}\] in form of \[{{i}^{2}}\] as \[{{\left( {{i}^{2}} \right)}^{17}}\] and it is equal to:
\[=\left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\times \left( {{i}^{2}} \right)\]
And we know that \[{{i}^{2}}=-1\]
So, it can be written as:
\[=\left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)\]
So, the solution of this is:
\[\begin{align}
  & =\left( 1 \right)\times \left( 1 \right)\times \left( 1 \right)\times \left( 1 \right)\times \left( 1 \right)\times \left( 1 \right)\times \left( 1 \right)\times \left( 1 \right)\times \left( -1 \right)\times \left( 1 \right)\times \left( -1 \right) \\
 & =-1 \\
\end{align}\]
So, the value of \[{{i}^{34}}\] is equal to -1.
During solving this type questions you should be change your given term in a form of \[{{\left( {{i}^{2}} \right)}^{n}}\] and then if the n is equal to any odd number then the value of that is -1 but if the n is equal to even number then the value of that term will be equal to 1.