Question

How do you simplify \[{i^{14}}?\]

Answer 1

Hint: We will use the formula of imaginary number to simplify the given question.
On doing some simplification we get the required answer.

Formula Used:
In mathematics, if any negative number is written under the square root then it is called an imaginary number.
The imaginary unit number is called as complex numbers, where \[i\] is defined as imaginary or unit imaginary.
Suppose we want to calculate roots for the equation \[{x^2} = 1\] , there we can find two different real roots of \[x\] .
But if we want to find roots of \[x\] in the equation \[{x^3} = 1\] , there we can have three different roots for \[x\] .
So, to solve it, we can perform following steps:
\[{x^3} = 1\]
\[ \Rightarrow {x^3} - 1 = 0\]
\[ \Rightarrow (x - 1)({x^2} + x + 1) = 0\]
So, it is true for the above equation that:
Either \[(x - 1) = 0\] or \[({x^2} + x + 1) = 0\] .
So, \[x\] can have a value of \[1\] .
And, if we solve \[({x^2} + x + 1) = 0\] , we can find two roots of \[x\] also.
So,
\[ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {{1^2} - 4 \times 1 \times 1} }}{{2 \times 1}} = \dfrac{{ - 1 \pm \sqrt { - 3} }}{2} = \dfrac{{ - 1 \pm \sqrt 3 i}}{2}.\]
So, the number \[ - 1\] , under the square root is called an imaginary unit and this kind of roots or numbers are called complex numbers.
So, the quantity of \[i\] is \[\sqrt { - 1} \] .
Or we can write it as \[i = \sqrt { - 1} \] .
So, it is obvious that if we multiply \[i\] even numbers of times then it will give us a real number, but if we multiply \[i\] odd numbers of times then it will give us an imaginary number.

Complete step by step answer:
The given expression in the question is \[{i^{14}}\] .
We can re-write the power of \[i\] in following way:
\[{i^{14}} = {\left( {{i^2}} \right)^7}...............(1)\]
Now, we know that \[i = \sqrt { - 1} \] .
So, if we squared both the terms in \[i = \sqrt { - 1} \] , we get:
\[{i^2} = - 1\] .
So, we can re-write the equation \[(1)\] as following:
\[{i^{14}} = {\left( { - 1} \right)^7}\] .
Now, if we multiply \[ - 1\] even number of times it will give us \[1\] but if we multiply \[ - 1\] odd numbers of times it will give us \[ - 1\] .
Here, the power of \[ - 1\] is \[7\] , which is an odd number.
Therefore,
\[{i^{14}} = {\left( { - 1} \right)^7} = - 1\] .

The value of \[{i^{14}} = - 1\] .

Note: Points to remember:
A complex number is expressed as following:
 \[X + i.Y\] , where \[X\] and \[Y\] are real numbers but the imaginary part of the number is \[i\] .
A complex number lies on the imaginary axis in \[X - Y\] plane.