Write as a simple fraction in lowest terms

The first term in the numerator has a factor of [itex](x^2+ 4)^{1/2}[/itex] and the second a factor of [itex](x^2+ 4)^{-1/2}[/itex]. -1/2 is the smaller power so note that [itex](x^2+ 4)^{1/2}= (x^2+ 4)(x^2+ 4)^{-1/2}[/itex] and factor out [itex](x^2+ 4)^{-1/2}[/itex]. The first term has a factor of [itex](x^2- 9)^{-2/3}[/itex] and the second a factor of [itex](x^2- 9)^{1/3}[/itex]. -2/3 is the smaller power so note that [itex](x^2- 9)^{1/3}= (x^2- 9)(x^2- 9)^{-2/3}[/itex] and factor out [itex](x^2- 9)^{-2/3}[/itex]. Of course, there is an x in both terms so factor that out:
[tex]x(x^2+ 4)^{-1/2}(x^2- 9)^{-2/3}\frac{\frac{2}{3}(x^2+ 4)- x^2+ 9}{x^2+ 4}[/tex]
Of course that [itex]x^2+ 4[/itex] in the denominator can be absorbed into the [itex](x^2+ 4)^{-1/2}[/itex] to give [itex](x^2+ 4)^{-3/2}[/itex].

I don't do the tex "automatically" but the you can "edit" and manually remove the "i". Sometimes when I realize that I have used a number of "itex"s where I want "tex", I copy the whole thing to the "clipboard", open "Notepad" (standard with Windows), paste into Notepad, use the editing features there, then reverse.

 

You haven't made a mistake yet, because both answers are equivalent. Just expand the numerator and collect like terms.

Oh and you made a typo in the numerator, you forgot the square in x²-9 :

[tex]\frac{x(\frac{2}{3}(x^{2}+4)-(x^2-9))}{(x^{2}+4)^{3/2}(x^{2}-9)^{2/3}}[/tex]

 

Oh, well, np