Question

How do you simplify ${i^{100}}$?

Answer 1

In order to simplify the above value, first write its exponent in the form of $2n$, and use the property of exponent that ${a^{m \times n}} = {\left( {{a^n}} \right)^m}$ to rewrite the expression and put the value of ${i^2} = - 1$. Then ,again write the exponent in the form of $2n$, use the fact that the square of any negative number is always a positive number to obtain your required result.

Formula:
${(A + B)^2} = {A^2} + {B^2} + 2 \times A \times B$
$(A + B)(A - B) = {A^2} - {B^2}$

Complete step by step solution:
We are given an imaginary number known as ‘iota’ raised to the power 100 i.e. ${i^{100}}$
In order to simplify this, we have to convert the exponent value in the form of $2 \times n$.
As we can see $100$ can be written as $2 \times 50$, where $50$ is $n$.
So ,rewriting ${i^{100}}$,we get
$= {i^{2 \times 50}}$
Now using property of exponent that ${a^{m \times n}} = {\left( {{a^n}} \right)^m}$,we can write
$= {\left( {{i^2}} \right)^{50}}$
Putting the value of ${i^2} = - 1$in above equation, we get
$= {\left( { - 1} \right)^{50}}$
Now again splitting $50$ in the form of $2 \times n$,we can rewrite our expression as
$= {\left( {{{\left( { - 1} \right)}^2}} \right)^{25}}$
Since, square of any negative value is positive
$= {\left( 1 \right)^{25}} \\\ = 1 \\\$
Therefore, the simplification of ${i^{100}}$ is equal to $1$.
Additional Information:
1. Real Number: Any number which is available in a number system, for example, positive, negative, zero, whole number, discerning, unreasonable, parts, and so forth are Real numbers. For instance:
12, - 45, 0, 1/7, 2.8, √5, and so forth, are all the real numbers.
2. A Complex number is a number which are expressed in the form $a + ib$ where $ib$ is the imaginary part and $a$ is the real number .i is generally known by the name iota. $$$$
or in simple words complex numbers are the combination of a real number and an imaginary number .
3. Complex numbers are very useful in representing periodic motion like water waves, light waves and current and many more things which depend on sine or cosine waves.

Note:
1.The Addition or multiplication of any 2-conjugate complex number always gives an answer which is a real number.
2.Value of ${i^2}$is equal to $- 1$.
3. Value of ${i^3} = {i^2}.i = - i$.