Question

Simplify the following:

Answer 1

`(\left(12x-56\right)x)/(\left(x-4\right)\left(-x^2+20x-32\right))`

Explanation:

Simplify

`((16)/(x-2)-(4)/(x-4))/((16)/(x)-(x-4)/(x-2))`

The first-level numerator is

`(16)/(x-4)-(4)/(x-2)\\\\\\= (16(x-4) - 4(x - 2))/((x-2)(x-4))\\\\\\= (16x -64 - 4x + 8)/((x-2)(x-4))\\\\\\= (12x -56)/((x-2)(x-4))`

The first-level denominator is

`(16)/(x)-(x-4)/(x-2)\\\\\\= (16(x-2) - x(x-4))/(x(x-2))\\\\\\\\= (16x - 32 -x^2 +4x)/(x(x-2))`

`= (-x^2+20x-32)/(x\left(x-2\right))`

Therefore

`((16)/(x-2)-(4)/(x-4))/((16)/(x)-(x-4)/(x-2))\\\\\\= (12x -56)/((x-2)(x-4)) /(-x^2+20x-32)/(x\left(x-2\right)) \\\\\\`

Use the fraction rule:
`(a)/(b) / (d)/(c) = (a)/(b) \cdot (d)/(c)`

`= (12x -56)/((x-2)(x-4)) \cdot (x(x-2))/(-x^2 +20x -32)}`

The (x-2) term cancels out resulting in:

`(\left(12x-56\right)x)/(\left(x-4\right)\left(-x^2+20x-32\right))`