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Question: How do you simplify \({i^{100}}\)?...

How do you simplify i100{i^{100}}?

Explanation

Solution

In order to simplify the above value, first write its exponent in the form of 2n2n, and use the property of exponent that am×n=(an)m{a^{m \times n}} = {\left( {{a^n}} \right)^m} to rewrite the expression and put the value of i2=1{i^2} = - 1. Then ,again write the exponent in the form of 2n2n, use the fact that the square of any negative number is always a positive number to obtain your required result.

Formula:
(A+B)2=A2+B2+2×A×B{(A + B)^2} = {A^2} + {B^2} + 2 \times A \times B
(A+B)(AB)=A2B2(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. i100{i^{100}}
In order to simplify this, we have to convert the exponent value in the form of 2×n2 \times n.
As we can see 100100 can be written as 2×502 \times 50, where 5050 is nn.
So ,rewriting i100{i^{100}},we get
=i2×50= {i^{2 \times 50}}
Now using property of exponent that am×n=(an)m{a^{m \times n}} = {\left( {{a^n}} \right)^m},we can write
=(i2)50= {\left( {{i^2}} \right)^{50}}
Putting the value of i2=1{i^2} = - 1in above equation, we get
=(1)50= {\left( { - 1} \right)^{50}}
Now again splitting 5050 in the form of 2×n2 \times n,we can rewrite our expression as
=((1)2)25= {\left( {{{\left( { - 1} \right)}^2}} \right)^{25}}
Since, square of any negative value is positive
=(1)25 =1  = {\left( 1 \right)^{25}} \\\ = 1 \\\
Therefore, the simplification of i100{i^{100}} is equal to 11.
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+iba + ib where ibib is the imaginary part and aa 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 i2{i^2}is equal to 1 - 1.
3. Value of i3=i2.i=i{i^3} = {i^2}.i = - i.