How can I simplify this algebraic expression?

[nombreuso]

So I came upon the expression -x^-1/3 (1-x^2/3)^1/2, that when simplified is -(x^-2/3-1)^1/2
I have tried many times but I din't know how to simplify it. ¿How can I do it?

 

[HallsofIvy]

Since \(\displaystyle (x^2)^{1/2}= x\) (I'm assuming x is positive), going the other way, \(\displaystyle x(anything)^{1/2}= (x^2(anything))^{1/2}\). In particular \(\displaystyle -x^{-1/3}(1- x^{2/3})^{1/2}= -(x^{-2/3}(1-x^{2/3}))^{1/2}= -(x^{-2/3}- x^{-2/3+ 2/3})^{1/2}= -(x^{-2/3}- x^0)^{1/2}= -(x^{-2/3}- 1)^{1/2}\).

 

[nombreuso]

Right, I forgot about that

 

[Deleted member 4993]

So I came upon the expression -x^-1/3 (1-x^2/3)^1/2, that when simplified is -(x^-2/3-1)^1/2
I have tried many times but I din't know how to simplify it. ¿How can I do it?

Same as response #2 - but sightly different representation:

\(\displaystyle -x^{-\frac{1}{3}} * \left( 1 - x^{\frac{2}{3}}\right) ^{\frac{1}{2}}\)............ assuming x \(\displaystyle \gt 0\)

\(\displaystyle = -\left( x^{-\frac{2}{3}}\right)^{\frac{1}{2}} * \left( 1 - x^{\frac{2}{3}}\right) ^{\frac{1}{2}}\)

\(\displaystyle = -\left( x^{-\frac{2}{3}} - 1 \right) ^{\frac{1}{2}}\)