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` .
Let `1=x/x` to be able to combine similar fractions.
Flip the fraction at the bottom to proceed to multiplication.
Apply `x/(x^(-1)+1)=x^2/(1+x)` , we get:
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)` .
Cancel out common factors to get rid of the denominators.
Apply distribution property.
The complex fraction `(1/x-x/(x^(-1)+1))/(5/x)` simplifies to `(1+x-x^3)/(5+5x)`