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Question: Find the value of \(\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}}\)...

Find the value of i6+i7+i8+i9i2+i3\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}}

Explanation

Solution

We have given an expression in ‘ii ’ where ‘ii’ is called IOTO. ‘ii’ is used to express the complex numbers. We know that value of ii = −1\sqrt { - 1} . So to solve this expression we put the value of ‘ii’. Before this, we factorize the expression and cancel the terms common in numerator and denominator. Then we put the value of i,i2,i3,i4i,{i^2},{i^3},{i^4} and solve it.

Complete step-by-step answer:
We have given
i6+i7+i8+i9i2+i3\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}}
The denominator is ii i2+i2{i^2} + {i^2}it can be written asi2(i+1){i^2}(i + 1).
The numerator is i6+i7+i8+i9{i^6} + {i^7} + {i^8} + {i^9}. It can be written as i2(i4+i5+i6+i7){i^2}({i^4} + {i^5} + {i^6} + {i^7}).
So i6+i7+i8+i9i2+i3=i2(i4+i5+i6+i7)i2(1+i)\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}} = \dfrac{{{i^2}({i^4} + {i^5} + {i^6} + {i^7})}}{{{i^2}(1 + i)}}
=(i4+i5+i6+i7)1+i\dfrac{{({i^4} + {i^5} + {i^6} + {i^7})}}{{1 + i}}
Again numerator is i4+i5+i6+i7{i^4} + {i^5} + {i^6} + {i^7}
It can be written as i4(1+i)+i6(1+i){i^4}(1 + i) + {i^6}(1 + i)taking (1+i)(1 + i)common, we get(i4+i6)(1+i)({i^4} + {i^6})(1 + i)
So the expression can be written as
i6+i7+i8+i9i2+i3=(i4+i6)(1+i)(1+i)=i4+i6\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}} = \dfrac{{({i^4} + {i^6})(1 + i)}}{{(1 + i)}} = {i^4} + {i^6}
As we know that i=−1i = \sqrt { - 1}
So i2=i×i=−1×−1=(−1)2=−1{i^2} = i \times i = \sqrt { - 1} \times \sqrt { - 1} = {\left( {\sqrt { - 1} } \right)^2} = - 1
So i4=i2×i2=(−1)×(−1)=1{i^4} = {i^2} \times {i^2} = ( - 1) \times ( - 1) = 1
i6=i4×i2=1×(−1)=−1{i^6} = {i^4} \times {i^2} = 1 \times ( - 1) = - 1
Therefore i4+i6=1+(−1)=1−1=0{i^4} + {i^6} = 1 + ( - 1) = 1 - 1 = 0
Therefore i6+i7+i8+i9i2+i3=0\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}} = 0.

Note: The complex number in mathematics are those numbers which can be written in a+iba + ib the form where ‘ii’ is the imaginary number called iota and has the value −1\sqrt { - 1} e.g. 2+3i2 + 3i is a complex number where 22 is its real part and 3i3i is its imaginary part. The combination of both real and imaginary parts is called a complex number. The main application of these numbers is that they represent periodic motion such as water waves alternating current light waves etc.
There are four types of algebraic expressions that we can apply to a complex number. These operations are addition, subtraction, multiplication, and division.