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Question

Question: How do you simplify \({{i}^{25}}\) ?...

How do you simplify i25{{i}^{25}} ?

Explanation

Solution

For answering this question we will use the basic concept of complex number which is given as i=1i=\sqrt{-1} and the basic formulae i4q+r=ir{{i}^{4q+r}}={{i}^{r}}. Now by applying these concepts we will come to a conclusion for this question.

Complete step by step answer:
Now considering from the question we have to find the value of i25{{i}^{25}} .
From the basic concept of complex numbers we know that i=1i=\sqrt{-1} and from these have one basic formulae i4q+r=ir{{i}^{4q+r}}={{i}^{r}} this can be derived by substituting values in place of qq and rr . Let us do that. By substituting r=0r=0 and q=1q=1 we will have i4=(1)4{{i}^{4}}={{\left( \sqrt{-1} \right)}^{4}} which can be further simplified as i4=(1)2=1{{i}^{4}}={{\left( -1 \right)}^{2}}=1 which is equal to i0=1{{i}^{0}}=1 .
Now we are going to use this formula in our question we can write 25=4(6)+125=4\left( 6 \right)+1 and we can also say that i25=i4(6)+1{{i}^{25}}={{i}^{4\left( 6 \right)+1}} by comparing it with the formulae we observe that q=6q=6 and r=1r=1 .
So we will have i25=i4(6)+1=i{{i}^{25}}={{i}^{4\left( 6 \right)+1}}=i .

Hence we can conclude that the value of i25{{i}^{25}} is equal to ii that is 1\sqrt{-1} .

Note: We should be sure with our calculations and concepts while answering questions of this type. In complex numbers we have many other concepts like these which we can prove similarly as shown here.