Solveeit Logo

Question

Question: How do you simplify \({{i}^{102}}\) ?...

How do you simplify i102{{i}^{102}} ?

Explanation

Solution

In the given expression we have ii . It is used to denote complex numbers on an imaginary plane. Whenever we get a negative value under the roots, we use this ii to remove the negation and further solve it. Use the identity associated with this ii 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{{i}^{102}}
The ii 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 ii to remove the negation and solve it further considering the values in the imaginary plane.
The complex number z=x+iyz=x+iy can be represented on the plane as the coordinates, (x,y)\left( x,y \right)
Given that xx is the real part of the complex number and yy is the imaginary part of the complex number.
The major definition of ii is given by,
i=1i=\sqrt{-1} or i2=1{{i}^{2}}=-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\Rightarrow {{i}^{102}}
We need to get i2{{i}^{2}} to easily place the value as -1 in the expression for that,
We shall split the exponent in the given below way.
(i2)51\Rightarrow {{\left( {{i}^{2}} \right)}^{51}}
Now let us substitute the value of i2{{i}^{2}} as -1 in the expression.
(1)51\Rightarrow {{\left( -1 \right)}^{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\Rightarrow -1

Hence the value of i102{{i}^{102}} is equal to the value, 1-1

Note: Any complex number can be denoted as z=x+iyz=x+iy and can be represented on the plane as the coordinates, (x,y)\left( x,y \right) . It is given that xx is the real part of the complex number and yy is the imaginary part of the complex number.