# Thread: length of a curve

1. ## length of a curve

Hi, i am trying to find the arc length of a curve over y=x^(2/3) from x=1 to x=8. I know that in order to use the length equation i have to find 1+[f '(x)]^2 which equals 1+(4/9)x^(-2/3). The next step in my book turns that into (9x^(2/3)+4)/(9x(2/3)). I cant figure out why they did that, can anyone help me.
thanks

2. Originally Posted by cowboys111
Hi, i am trying to find the arc length of a curve over y=x^(2/3) from x=1 to x=8. I know that in order to use the length equation i have to find 1+[f '(x)]^2 which equals 1+(4/9)x^(-2/3). The next step in my book turns that into (9x^(2/3)+4)/(9x(2/3)). I cant figure out why they did that, can anyone help me.
thanks
$\displaystyle 1 + \frac{4}{9} x^{-2/3}$

$\displaystyle = 1 + \frac{4}{9} \frac{1}{x^{2/3}}$

since $\displaystyle x^{-2/3} = \frac{1}{x^{2/3}}$. Get a common denominator:

$\displaystyle = \frac{9 x^{2/3}}{9 x^{2/3}} + \frac{4}{9 x^{2/3}}$

which turns into what the book has once you add the two fractions together.