Question
Question: How do you simplify \({i^{18}}\)?...
How do you simplify i18?
Solution
Let us first know what a complex number is. Complex numbers are any number in the form of Z=a+ib where a and b are real numbers and i is an imaginary number known as iota. It is equal to −1. Complex numbers are denoted as Z.
It must be known that when i is raised to certain powers, it assumes some values which are given as:
i=−1
i2=−1 (As i2=(−1)2=−1)
i3=−i (As i3=i2×i=−1×i=−i)
i4=1 (As i4=i2×i2=−1×−1=1)
We will use these 4 values of i to solve the given question.
Complete step by step solution:
Given expression is i18. We will now simplify it by breaking the exponent 18into factors, such that
⇒i18=i2×3×3
We will now separate the factors of the exponent into 2 and all other factors, such that
⇒i18=i2×9
We will now use one of the law of exponents which states xp×q=(xp)q, such that
⇒i18=(i2)9
On substituting the value of i2=−1, we will get
⇒i18=(−1)9
It must be known that if −1 is raised to the power of an odd number, we will get −1 itself as the answer. Since 9 is an odd number, therefore we will get
⇒i18=−1
Hence, on simplifying i18, we get −1 as the answer.
Note:
The given question can also be solved in an alternate way. We know that i4=1. So we will simplify the exponent 18of the expression i18 in the form of 4m+n where m and n are whole numbers. Hence on simplifying, we will get
⇒i18=i4×4+2
We will now use one of the law of exponents which states xp+q=xp×xq, such that
⇒i18=i4×4×i2
Again we will use one of the laws of exponents which states xp×q=(xp)q, such that
⇒i18=(i4)4×i2
We know that i4=1. Therefore on substituting this value in the above equation, we will get
⇒i18=(1)4×i2
⇒i18=1×i2
So i18 gets reduced as given,
⇒i18=i2
Now we know that i2=−1. Therefore on substituting this value in the above equation, we will get
⇒i18=−1
Hence on simplifying i18, we got −1.