<s>[INST] How to simplify this fraction with different powers? I happen to be stuck trying to simplify this:
$$\left[\frac{(3x+2)(x+1)^\frac{3}{2}-(\frac{3}{2}x^2+2x)(\frac{3}{2})(x+1)^\frac{1}{2}}{(x+1)^3}\right]$$
here's the simplified solution that I'm trying to figure out how it was reached
 [/INST] We have
$$\frac{(3x+2)(x+1)^\frac{3}{2}-(\frac{3}{2}x^2+2x)(\frac{3}{2})(x+1)^\frac{1}{2}}{(x+1)^3}$$
divide numerator and denominator by $(x+1)^{\frac12}$ giving
$$\frac{(3x+2)(x+1)-(\frac{3}{2}x^2+2x)(\frac{3}{2})}{(x+1)^\frac52}$$
multiply the $(3x+2)(x+1)$ to get $3x^2 + 5x + 2$ and the $(\frac32x^2 + 2x)(\frac32)$ to get $\frac94x^2 + 3x$ giving:
$$\frac{3x^2 + 5x + 2-\frac94x^2 - 3x}{(x+1)^\frac52}$$
then simplify the numerator:
$$3x^2 + 5x + 2-\frac94x^2 - 3x = \frac{12}{4}x^2 - \frac94x^2 + 5x - 3x + 2 = \frac34x^2 + 2x + 2$$
to get
$$\frac{\frac34x^2 + 2x + 2}{(x+1)^\frac52}$$
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