Question
Question: How do you simplify \({i^{37}}\)?...
How do you simplify i37?
Solution
In the question above we have a value of an imaginary number that states that it is with the power, i37, and we have to simplify it. While simplifying it, one must know that for an imaginary number, the value of its square is always −1.
This means that when multiplied with its fourth power, the number will be equal to 1. Using this conclusion in mind, we will solve this question, following the steps and showing required proof.
Complete step-by-step solution:
We have an imaginary number i37 and we have to find its value, for that we will start by recalling that,
⇒i=−1
Now, we know that the value of the imaginary number is square root of −1, using this for further calculation we get,
⇒i2=−1
Therefore, the value further will be,
⇒i3=−1×−1
And finally, the fourth power will hold the value like,
⇒i4=−1×−1
This will be of the value,
⇒i4=1
Whenever the exponent on i is a multiple of 4, it will evaluate to 1, as imaginary numbers follow a pattern.
36 is a multiple of 4, so we know i36=1. We can rewrite i37 as,
⇒i36×i
This can be substituted as,
⇒1×i
This will multiply to,
⇒i
Thus, i37=i
Therefore, the value of the imaginary number i37=i
Note: An imaginary number is a Complex number that can be written as a real number multiplied by the imaginary unit I, which is defined by its property i2=−1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary. The set of imaginary numbers is sometimes denoted using the blackboard bold letter.