(1/x-x/(x^(-1)+1))/(5/x) Simplify the complex fraction.

To evaluate the given complex fraction  (1/x-x/(x^(-1)+1))/(5/x) , we may simplify first the part x/(x^(-1)+1) .
Apply Law of Exponent: x^(-n)=1/x^n .
Let x^(-1)= 1/x^1 or  1/x .
x/(1/x+1)
Let 1=x/x to be able to combine similar fractions.
x/(1/x+x/x)
x/((1+x)/x)
Flip the fraction at the bottom to proceed to multiplication.
x*x/(1+x)
x^2/(1+x)
Apply x/(x^(-1)+1)=x^2/(1+x) , we get:
(1/x-x/(x^(-1)+1))/(5/x)
(1/x-x^2/(1+x))/(5/x)
Determine the LCD or least common denominator.
The denominators are x and (1+x) . Both are distinct factors.
Thus, we get the LCD by getting the product of the distinct factors from denominator side of each term.
LCD =x*(1+x) or x+x^2
Maintain the factored form of the LCD for easier cancellation of common factors on each term.
Multiply each term by the LCD=x*(1+x) .
(1/x*x*(1+x)-x^2/(1+x)*x*(1+x))/((5/x)x*(1+x))
Cancel out common factors to get rid of the denominators.
(1*(1+x)-x^2*x)/(5*(1+x))
Apply distribution property.
(1+x-x^3)/(5+5x)
or -(x^3-1-x)/(5x+5)
The complex fraction (1/x-x/(x^(-1)+1))/(5/x) simplifies to (1+x-x^3)/(5+5x) 

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