Question

How do you simplify \[{{i}^{752}}\]?

Answer 1

Hint: In this problem, we have to simplify the given imaginary part. We should know that any negative terms inside the square root is equal to an imaginary part, which will be a complex number. We can write it as \[i=\sqrt{-1}\]. Using this we can find the value of \[{{i}^{4}}\]. We can then split the given power term in terms of multiplication of 4 to find the value of the given imaginary part.

Complete step by step solution:
We know that the given imaginary part to be simplified is \[{{i}^{752}}\].
We know that any negative terms inside the square root is equal to an imaginary part, which will be a complex number.
We can write it as,
 \[i=\sqrt{-1}\].
We can now write the square term of \[i\], we get
\[\Rightarrow {{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]
 We can now write the fourth power of \[i\], using the above step, we get
\[\begin{align}
  & \Rightarrow {{\left( {{i}^{2}} \right)}^{2}}={{\left( -1 \right)}^{2}} \\
 & \Rightarrow {{i}^{4}}=1 \\
\end{align}\]
Now we can write the given power as,
\[\Rightarrow 752=188\times 4\]
We can write the given imaginary part as,
\[\Rightarrow {{i}^{752}}={{\left( {{i}^{4}} \right)}^{188}}\]
We can now substitute \[{{i}^{4}}=1\], we get
\[\Rightarrow {{i}^{752}}={{1}^{188}}=1\]
 Since anything to the power 1 is one itself.
Therefore, the value of \[{{i}^{752}}\] is 1.

Note: Students make mistakes while finding the value of the fourth power of the imaginary part \[i\]. We should always remember that the complex number exists when we have a negative term inside the root, where the imaginary part occurs. We should also remember that anything to the power one is one itself.