Graphically, plotting LHS and RHS separately shows an intersection at (2,64).

And plugging x=2 clearly shows it is a solution.

https://www.desmos.com/calculator/wdubnrojmz

Hand waving argument that there is only one solution:

The derivative of the LHS is always greater than or equal to 0, so LHS is strictly increasing.

The RHS is an even function which is strictly increasing for x>0.

The RHS minimum value occurs when x=0 and which evaluates to about 42.5 when x=0.

THe LHS is about 42.5 when x is about 1.95 so we can ignore all x values less than 1.95

Above x=2, the LHS is increasing approaching x^9 whereas the RHS is increasing approaching x^(8/3), so it is unlikely they will intersect a second time.