`(16/(x-2))/(4/(x+1)+6/x)` Simplify the complex fraction.

Expert Answers
marizi eNotes educator| Certified Educator

To simplify the given complex fraction `(16/(x-2))/(4/(x+1)+6/x)` , we may look for the LCD or least common denominator.

The denominators are `(x-2)` , `x` , and `(x+1)` . All are distinct factors.

Thus, we get the LCD by getting the product of the distinct factors from denominator side of each term.

`LCD =(x-2)*x* (x+1)`

Maintain the factored form of the LCD for easier cancellation of common factors on each term.

Multiply each term by the LCD=(x-2)*x* (x+1).

`(16/(x-2)*(x-2)*x* (x+1))/(4/(x+1)*(x-2)*x* (x+1)+6/x*(x-2)*x* (x+1))`

`(16*x* (x+1))/(4*(x-2)*x +6*(x-2)* (x+1))`

Apply distributive property.

`(16x*(x+1))/((4x-8)*x +(6x-12)* (x+1))`

`(16x^2+16x)/((4x^2-8x) +(6x^2+6x-12x-12))`

Combine possible like terms.

`(16x^2+16x)/((4x^2-8x) +(6x^2-6x-12))`

`(16x^2+16x)/(4x^2-8x+6x^2-6x-12)`

`(16x^2+16x)/(10x^2-14x-12)`

Factor out 2 from each side.

`(2(8x^2+8x))/(2(5x^2-7x-6))`

Cancel out common factor `2` .

`(8x^2+8x)/(5x^2-7x-6)`

 The complex fraction `(16/(x-2))/(4/(x+1)+6/x)` simplifies to `(8x^2+8x)/(5x^2-7x-6)` .