Write as a simple fraction in lowest terms

[mindauggas]

1. The problem statement, all variables and given/known data

\frac{\frac{2}{3}x(x^{2}+4)^{1/2}(x^{2}-9)^{-2/3}-x(x^{2}-9)^{1/3}(x^{2}+4)^{-1/2}}{x^{2}+4}

3. The attempt at a solution

\frac{x(x^{2}+4)^{-1/2}(x^{2}-9)^{-2/3}(\frac{2}{3}(x^{2}+4)-(x-9))}{x^{2}+4}

Then:

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

What should I do next? I multiply the numerator but this, it seems leads to a dead-end. Or is there a mistake involved in the aforementioned steps?

P. S. The book gives the answer

\frac{-x^{3}+35x}{3(x^{2}+4)^{3/2}(x^{2}-9)^{2/3}}

P. S. S. Can someone tell me how to write tex instead of itex automatically?

[HallsofIvy]

1. The problem statement, all variables and given/known data

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

3. The attempt at a solution

\frac{x(x^{2}+4)^{-1/2}(x^{2}-9)^{-2/3}(\frac{2}{3}(x^{2}+4)-(x-9))}{x^{2}+4}

Then:

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

What should I do next? I multiply the numerator but this, it seems leads to a dead-end. Or is there a mistake involved in the aforementioned steps?

P. S. The book gives the answer

\frac{-x^{3}+35x}{3(x^{2}+4)^{3/2}(x^{2}-9)^{2/3}}

P. S. S. Can someone tell me how to write tex instead of itex automatically?
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.

[mindauggas]

I don't understand, you just rewrote what I did, or have I overlooked something?

[Mentallic]

Then:

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

What should I do next? I multiply the numerator but this, it seems leads to a dead-end. Or is there a mistake involved in the aforementioned steps?

P. S. The book gives the answer

\frac{-x^{3}+35x}{3(x^{2}+4)^{3/2}(x^{2}-9)^{2/3}}

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 :

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

[mindauggas]

The typo was the mistake (as usual for me). Thank's for helping...

[Mentallic]

The typo was the mistake (as usual for me). Thank's for helping...

Oh, well, np :biggrin: