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Question

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

How do you simplify i752{{i}^{752}}?

Explanation

Solution

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=1i=\sqrt{-1}. Using this we can find the value of i4{{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 i752{{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=1i=\sqrt{-1}.
We can now write the square term of ii, we get
i2=(1)2=1\Rightarrow {{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1
We can now write the fourth power of ii, 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.