Question

How do you simplify ${{i}^{752}}$?

Answer 1

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

& \Rightarrow {{\left( {{i}^{2}} \right)}^{2}}={{\left( -1 \right)}^{2}} \\\ & \Rightarrow {{i}^{4}}=1 \\\ \end{aligned}$$ Now we can write the given power as, $$\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.