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Question: Find the value of \(\dfrac{{{i^{592}} + {i^{590}} + {i^{588}} + {i^{586}} + {i^{584}}}}{{{i^{582}} +...

Find the value of i592+i590+i588+i586+i584i582+i580+i578+i576+i574−1\dfrac{{{i^{592}} + {i^{590}} + {i^{588}} + {i^{586}} + {i^{584}}}}{{{i^{582}} + {i^{580}} + {i^{578}} + {i^{576}} + {i^{574}}}} - 1.

Explanation

Solution

We have to solve a problem of complex numbers and the letter I in the question is called iota which is having a certain value and we have to use the properties of iota to solve the question. We don’t have to solve these high powers of iota in the question but we have to use the properties.

Complete Step by Step Solution:
According to the question, we have to calculate the value of i592+i590+i588+i586+i584i582+i580+i578+i576+i574−1\dfrac{{{i^{592}} + {i^{590}} + {i^{588}} + {i^{586}} + {i^{584}}}}{{{i^{582}} + {i^{580}} + {i^{578}} + {i^{576}} + {i^{574}}}} - 1
Here ii is iota which is equal to −1\sqrt { - 1}
So we will use the formulas of complex numbers that is when we raise iota with any power and its results
When we square the iota we will get⇒i2=−1 \Rightarrow {i^2} = - 1
When we do cube of the iota we will get⇒i3=−i \Rightarrow {i^3} = - i
When we do power equal to four of the iota we will get⇒i4=1 \Rightarrow {i^4} = 1
So we can convert the question into new form by using the above knowledge, as
⇒(i4)148+(i4)147(i2)+(i4)147+(i4)146(i2)+(i4)146(i4)145(i2)+(i4)145+(i4)144(i2)+(i4)144+(i4)143(i2)−1\Rightarrow \dfrac{{{{({i^4})}^{148}} + {{({i^4})}^{147}}({i^2}) + {{({i^4})}^{147}} + {{({i^4})}^{146}}({i^2}) + {{({i^4})}^{146}}}}{{{{({i^4})}^{145}}({i^2}) + {{({i^4})}^{145}} + {{({i^4})}^{144}}({i^2}) + {{({i^4})}^{144}} + {{({i^4})}^{143}}({i^2})}} - 1
Now we can simplify the above equation as
⇒1−1+1−1+1−1+1−1+1−1−1\Rightarrow \dfrac{{1 - 1 + 1 - 1 + 1}}{{ - 1 + 1 - 1 + 1 - 1}} - 1

⇒−1−1=−2 \Rightarrow - 1 - 1 = - 2
Hence this is our answer.

Note:
Iota is an imaginary number which is equal to −1\sqrt { - 1} .We have to notice that the calculation is long in this question so we have to pay concentration while solving the question. We can also generalize the values of iota for different values of its power and can solve the question in a different way, that is, by using the generalized formulas of the equation. A complex number is represented in the form of a+iba + ib, where aa is the real part and ibib is the imaginary part.