Question

I need the answer to both please. An answer plus an explanation would be even better.

Answer 1

2. 1

3.
`(x(x + 2)(x - 4))/((x - 2)^2)`

Explanation:

2.

`(x^2 - 81)/(x + 81) * (x^2 + 81x)/(x^3 - 81x) =`

Multiply the fractions together by multiplying the numerators and multiplying the denominators.

`= ((x^2 - 81)(x^2 + 81x))/((x + 81)(x^3 - 81x))`

Factor every numerator and denominator.

`= ((x + 9)(x - 9)x(x + 81))/((x + 81)x(x^2 - 81))`

`= ((x + 9)(x - 9)x(x + 81))/((x + 81)x(x + 9)(x - 9))`

Now divide the numerator and denominator by terms common to both. This is what is commonly called canceling terms in the numerator and denominator. Every term in the numerator has an equal term in the denominator. All terms cancel out leaving 1.

`= 1`

3.

Since you have a division here, first, multiply the first fraction by the reciprocal of the second fraction. Then factor the numerator and denominator and cancel out common terms.

`(x^2 + 4x)/(x - 2) / (x^2 + 2x - 8)/(x^2 - 2x - 8) =`

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

`= (x^2 + 4x)/(x - 2) * (x^2 - 2x - 8)/(x^2 + 2x - 8)`

`= ((x^2 + 4x)(x^2 - 2x - 8))/((x - 2)(x^2 + 2x - 8))`

Now factor every factorable expression.

`= (x(x + 4)(x + 2)(x - 4))/((x - 2)(x + 4)(x - 2))`

Now cancel equal terms in the numerator and denominator.

`= (x(x + 2)(x - 4))/((x - 2)(x - 2))`

`= (x(x + 2)(x - 4))/((x - 2)^2)`