solve for x. X^(5/3) - 6X = X^(4/3)

Question

The question is asking to solve for X.

Answer 1

x^5/3-6x-x^4/3=0.

Let y³=x, then y⁵-6y³-y⁴=0=y³(y²-y-6)=y³(y-3)(y+2), so y=0, 3 or -2.

Therefore x=0, 27, -8.

CHECK

x=0 satisfies the original equation.

When x=27, x^5/3=3⁵=243; x^4/3=3⁴=81; 6x=162.

x^5/3-6x=243-162=81=x^4/3.

When x=-8, x^5/3=(-2)⁵=-32; x^4/3=(-2)⁴=16; 6x=-48.

x^5/3-6x=-32+48=16=x^4/3.

So solution is confirmed.