Question
Question: How do you simplify \({i^{100}}\)?...
How do you simplify i100?
Solution
In order to simplify the above value, first write its exponent in the form of 2n, and use the property of exponent that am×n=(an)m to rewrite the expression and put the value of i2=−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=A2+B2+2×A×B
(A+B)(A−B)=A2−B2
Complete step by step solution:
We are given an imaginary number known as ‘iota’ raised to the power 100 i.e. i100
In order to simplify this, we have to convert the exponent value in the form of 2×n.
As we can see 100 can be written as 2×50, where 50 is n.
So ,rewriting i100,we get
=i2×50
Now using property of exponent that am×n=(an)m,we can write
=(i2)50
Putting the value of i2=−1in above equation, we get
=(−1)50
Now again splitting 50 in the form of 2×n,we can rewrite our expression as
=((−1)2)25
Since, square of any negative value is positive
=(1)25 =1
Therefore, the simplification of i100 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 i2is equal to −1.
3. Value of i3=i2.i=−i.