Question

Simplify and show work:

[(X^(2)+3)^(1/3)](-1)(x^(-2))+[1/3(x^(2)+3)^(-2/3)](2x)(x^(-1))


My answer key says that it's this but I have no idea how to get to it. I've tried 4 times already and failed each:

[-(x^(2)+9)]/[(3x^(2))(x^(2)+3))^(2/3)]

Answer 1

The answer makes me think you are trying to find the derivative of


∛(x^2+3)/x = ∛(x^2+3) * x^-1

That would be

∛(x^2+3)*(-1)(x^-2) + (1/3)(x^2+3)^(-2/3) * (2x)(1/x)

To put all that over (x^2+3)^(2/3), multiply the first term, and you get

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

=

-(x^2+3) + (2/3)x^2
-------------------------
x^2(x^2+3)^(2/3)

=

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

=
-(x^2+9)
------------------------------
3x^2(x^2+3)^(2/3)