Write as a simple fraction in lowest terms

[mindauggas]

Homework Statement

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

The 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]

Homework Statement

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

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 $(x^2+ 4)^{-3/2}$.

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