Simplify the rational expression

3 (1 + x)^1/3 - x (1 + x)^-2/3 divided by (1 + x)^2/3

Correction: the question is to simplify the FRACTIONAL expression.

just to unclutter things, let u = (1+x)^1/3. Then we have

(3u-x/u^2)/u^2
= (3u^3-x)/u^4
= (3(1+x)-x)/(1+x)^4/3
= (2x+3)/(1+x)^4/3

To simplify the rational expression, we need to simplify the numerator and the denominator separately, and then divide the simplified numerator by the simplified denominator.

Let's start with the numerator: 3(1 + x)^(1/3) - x(1 + x)^(-2/3).

1. Distribute the exponent of 1/3 to both terms in the numerator:
3(1 + x)^(1/3) = 3^(1/3)(1 + x)^(1/3)

2. Distribute the exponent of -2/3 to the term x:
x(1 + x)^(-2/3) = x(1 + x)^(-2/3) = x(-2/3)(1 + x)^(-2/3-1/3)

Simplifying the exponent:
x(-2/3)(1 + x)^(-2/3-1/3) = -2x/3(1 + x)^(-1).

Now let's simplify the denominator: (1 + x)^(2/3).

To divide these two terms, we can multiply the numerator by the reciprocal of the denominator. So the simplified expression becomes:

[3^(1/3)(1 + x)^(1/3) - (2x/3)(1 + x)^(-1)] / (1 + x)^(2/3).

To further simplify this expression, we can use the exponent properties:
- For the numerator, since we have two terms with the same base (1 + x), we can combine them by adding the exponents.
- For the denominator, the same base (1 + x) is elevated to a positive exponent, so we use the exponent property and change it to the denominator.

After applying these properties, the expression becomes:

[3^(1/3)(1 + x)^(1/3 - 2/3) - (2x/3)(1 + x)^(2/3 - 2/3)] / (1 + x)^(2/3).

Simplifying the exponents:
[3^(1/3)(1 + x)^(-1/3) - (2x/3)(1 + x)^0] / (1 + x)^(2/3).

Simplifying the denominator as (1 + x)^(2/3):
[3^(1/3)(1 + x)^(-1/3) - (2x/3)] / (1 + x)^(2/3).

This is the simplified form of the rational expression.