Question

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

Answer 1

Hint: 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, ${{i}^{102}}$
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, $\left( x,y \right)$
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,
$i=\sqrt{-1}$ or ${{i}^{2}}=-1$
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.
$\Rightarrow {{i}^{102}}$
We need to get ${{i}^{2}}$ to easily place the value as -1 in the expression for that,
We shall split the exponent in the given below way.
$\Rightarrow {{\left( {{i}^{2}} \right)}^{51}}$
Now let us substitute the value of ${{i}^{2}}$ as -1 in the expression.
$\Rightarrow {{\left( -1 \right)}^{51}}$
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,
$\Rightarrow -1$

Hence the value of ${{i}^{102}}$ 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, $\left( x,y \right)$ . It is given that $x$ is the real part of the complex number and $y$ is the imaginary part of the complex number.