Question
Question: How do you simplify \({{i}^{59}}\)?...
How do you simplify i59?
Solution
To solve this question, we need to make the exponent on i equal to a multiple of four. Since the exponent is equal to 59 which is one less than 60, a multiple of four, we need to multiply and divide the given expression by i to obtain ii60. Then, we have to use the relation i4=1 to show that any multiple of four, raised to i given one. Using this our expression will get reduced to i1. Finally, on dividing and multiplying the obtained expression by i we will get the simplified expression.
Complete step by step solution:
Let us write the expression given in the above question in the below equation as
⇒E=i59........(i)
Now, we know that i is equal to the square root one minus one, which in turn means that the square of i is equal to minus one, that is,
⇒i2=−1.......(ii)
Squaring both the sides, we get
⇒(i2)2=(−1)2⇒i4=1
Raising the terms on both sides of the above equation to the exponent of n, where n is a natural number, we get
⇒(i4)n=(1)n⇒i4n=1
From the above equation, we can say that the value of i raised to a multiple of four is equal to one.
Now, we consider the equation (i)
⇒E=i59
Multiplying and dividing by i we get
⇒E=i59×ii⇒E=ii60
Since 60 is a multiple of four, we can substitute i60=1 in the above equation to get
⇒E=i1
Multiplying and dividing by i we get
⇒E=i2i
Finally, substituting (ii) in the above equation, we get
⇒E=−1i⇒E=−i
Hence, the given expression is simplified as −i.
Note: The equation i4n=1, which we obtained in the above solution is an identity. We must remember it in order to quickly solve these kinds of problems. Also, we must remember the identities i2=−1 and i1=−i must be remembered for solving these types of questions.