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

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marizi eNotes educator| Certified Educator

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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