Question

How do you simplify `[(x+4)^3/4(x+2)^-2/3 - (x+2)^1/3(x+4)^-1/4]/ [(x+4)^3/4]^2`?

Answer 1

`4/(3(x+4)^3) *[1/(x+2)^2 - (x+2)/(x+4)^4]`

Given:
`(((x+4)^3)/4 * (x+2)^-2/3 - (x+2)/3 * (x+4)^-1/4)/((x+4)^3/4)^2`

Let `a = x+4` and `b=x+2`
`((a^3)/4 * b^-2/3 - b/3 * a^-1/4)/(a^3/4)^2`

When you divide by a fraction, you are multiplying the reciprocal:
`[(a^3)/4 * b^-2/3 - b/3 * a^-1/4] * (4/a^3)^2`

Change negative exponents to reciprocals with positive exponents:
`[(a^3)/4 * 1/(3b^2) - b/3 * 1/(4a)] * (4/a^3)^2`

`[a^3/(12b^2) - b/(12a)] (16/a^6)`

`(16a^3)/(12a^6b^2) - (16b)/(12a^7)`

`4/(3a^3b^2) - (4b)/(3a^7)`

Factor out `4/(3a^3)`:

`4/(3a^3) [1/b^2 - b/a^4]`

Substitute back in `x+4` and `x+2`
`4/(3(x+4)^3) * [1/(x+2)^2 - (x+2)/(x+4)^4]`

Hope that was helpful.