1. ## Common Denominator ?

How do I do this?

(1/ 3x^(2/3)) + (2/3x^(5/3))

thanks

2. ## Re: Common Denominator ?

I guess this is the exercice:
$\frac{1}{3\cdot x^{\frac{2}{3}}}+\frac{2}{3\cdot x^{\frac{5}{3}}}$
?
Write:
$3x^{\frac{5}{3}}=3x^{\frac{2}{3}}.x^{\frac{3}{3}}= 3x^{\frac{2}{3}}.x$

Can you now finish it? ...

3. ## Re: Common Denominator ?

sorry but can you please clarify?

I still don't understand

4. ## Re: Common Denominator ?

Yes, but I edited my post, because there was a mistake in it.
I just used the basic rule:
$a^x\cdot a^y=a^{x+y}$

It also important, is it:
$(3x)^{\frac{5}{3}}$ or $3\cdot x^{\frac{5}{3}}$
I think the second one. Can you confirm this? ...

5. ## Re: Common Denominator ?

The original problem was taking the derivative of x^(1/3) - x^(-2/3)

so

(1/3)x^(1/3) - (-2/3)x^(-5/3)

Would I have to multiply (1/ 3x^(2/3)) by x^(3/3) to get it to make it the same denominator right?

6. ## Re: Common Denominator ?

You made a mistake in your derivative, but in your first post it was correct so I guess you didn't see it:
$D\left(x^{\frac{1}{3}}\right)=\frac{1}{3}x^{\frac{-2}{3}}$
So you get indeed:
$\frac{1}{3\cdot x^{\frac{2}{3}}}+\frac{2}{3\cdot x^{\frac{5}{3}}}=\frac{1}{3} \left(\frac{1}{x^{\frac{2}{3}}}+\frac{2}{x^{\frac{ 5}{3}}} \right)$
Now, you can wright, like I said before:
$x^{\frac{5}{3}}=x^{\frac{2}{3}}\cdot x$
So you get:
$\frac{1}{3} \left(\frac{1}{x^{\frac{2}{3}}}+\frac{2}{x^{\frac{ 2}{3}}\cdot x}\right)$
Now if you multiply the left fraction with $x$ in the numerator and denominator you'll get the same denominator as the right fraction, so:
$...=\frac{1}{3} \left(\frac{x}{x^{\frac{5}{3}}}+\frac{2}{x^{\frac{ 5}{3}}}\right)=\frac{2+x}{3\cdot x^{\frac{5}{3}}}$