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Express each of the complex number in the form a+ib. (3) i^-39

Express each of the complex number in the form a+ib.

Q : 3        i^{-39}

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We know that  i^4 = 1
Now, we will reduce   i^{-39}   into 

i^{-39} = (i^{4})^{-9}.i^{-3}
         = (1)^{-9}.(-i)^{-1}                                                     (\because i^4 = 1 , i^3 = -i)
         = \frac{1}{-i}
         = \frac{1}{-i} \times \frac{i}{i}
         = \frac{i}{-i^2}                                                                        (\because i^2 = -1)
         = \frac{i}{-(-1)}
         =i
Now, in the form of  a+ib  we can write it as
o+i1
Therefore, the answer is o+i1

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