Thread: How does this simplification work?

This is a calculus question, but I understand how the differentiation here works. What I don't understand is the simplification, which is generally algebra, so I posted it in this forum. I just can't figure out how they have done the simplification? I can get the end of the first line, but how they simplify it to the final outcome, can anyone help me understand? The second line I have no idea... how does -x^(3/5) suddenly become -5x/5x^(2/5)?

$\displaystyle x^{3/5}$ is multiplied and divided by $\displaystyle 5x^{2/5}$ to find the common denominator of the two terms.

To put the terms in the same fraction they need to have the same common denominator: $\displaystyle 5x^{2/5}$

So you must do:

$\displaystyle {-x^{3/5} = \frac{-x^{3/5} \cdot 5x^{2/5}}{5x^{2/5}} \ = \frac{-5x^{3/5 + 2/5}}{5x^{2/5}} \ = \frac{-5x^{5/5}}{5x^{2/5}} \ = \frac{-5x^1}{5x^{2/5}} \ = \frac{-5x}{5x^{2/5}}$

now as both terms in the equation have the same denominator then can be put into the same fraction:

$\displaystyle \frac{-5x}{5x^{2/5}} + \frac{3(4-x)}{5x^{2/5}} = \frac{-5x + 3(4-x)}{5x^{2/5}}$

which simplifies to:

$\displaystyle \frac{12-8x}{5x^{2/5}}$