you can rewrite your expression as

$3x^{7/3} + 2x^{2/5} - 6=0$

$3x^{35/15} + 2x^{6/15}-6 =0$

$u=x^{1/15}$

$3u^{35} + 2u^6 - 6 = 0$

This can be solved numerically by your root finding algorithm of choice. In this case there will be 35 solutions....

The one real root is $u \approx 1.00758$

That solution $u_0$ can then be used to find $x$

$x=u^{15}$

$x=(1.00758)^{15} \approx 1.11991$