Question
Question: How do you simplify \({{i}^{102}}\) ?...
How do you simplify i102 ?
Solution
In the given expression we have i . It is used to denote complex numbers on an imaginary plane. Whenever we get a negative value under the roots, we use this i to remove the negation and further solve it. Use the identity associated with this i and then rewrite the expression in such a way that the identity could be easily applied and then evaluated.
Complete step by step solution:
The given expression is, i102
The i used here is mainly used in the representation of the complex numbers.
Whenever we get a negative value under roots while solving for the roots or by any other way,
We use this i to remove the negation and solve it further considering the values in the imaginary plane.
The complex number z=x+iy can be represented on the plane as the coordinates, (x,y)
Given that x is the real part of the complex number and y is the imaginary part of the complex number.
The major definition of i is given by,
i=−1 or i2=−1
Now we use this to solve our question.
Firstly, we mold our expression in such a way that we can easily apply this above identity and solve it.
⇒i102
We need to get i2 to easily place the value as -1 in the expression for that,
We shall split the exponent in the given below way.
⇒(i2)51
Now let us substitute the value of i2 as -1 in the expression.
⇒(−1)51
Now we know that any number which is negative when raised to the power of an odd number results in a negative number.
Whereas any negative number which when raised to an even number, results in a positive number.
Since the power 51 is an odd number, we shall get the result as a negative number which is,
⇒−1
Hence the value of i102 is equal to the value, −1
Note: Any complex number can be denoted as z=x+iy and can be represented on the plane as the coordinates, (x,y) . It is given that x is the real part of the complex number and y is the imaginary part of the complex number.