The rules we use to manipulate powers are true in two situations: any base but integer exponent, or positive base with any exponent. When you have a negative (or complex) base and noninteger exponent you cannot trust the rules. One simple example is with the cubic root of a negative number. We expect (-8)^(1/3) to be -2. But then we cannot replace 1/3 with 2/6, because (-8)^(2/6)=+2. I know of a "proof" that 0=1, where the error is hard to single out, unless we are aware of the problems with powers.

the standard definition of fractional roots is to use the denominator first. So (-20)^(4/3) is the same as ((-20)^(1/3))^4. Below I show the outcomes of the two possible groupings and one can see which agrees with the ungrouped form.

In[571]:= (-20)^(4/3)

Out[571]= -20 (-5)^(1/3) 2^(2/3)

In[572]:= ((-20)^4)^(1/3)

Out[572]= 20 2^(2/3) 5^(1/3)

In[573]:= ((-20)^(1/3))^4

Out[573]= -20 (-5)^(1/3) 2^(2/3)

If you want the negative solution to show up you have to write the equation with Surd:

Solve[Surd[6 x - 5, 3]^4 == 625, x, Reals]

or with CubeRoot

Solve[CubeRoot[6 x - 5]^4 == 625, x, Reals]

This is because Mathematica powers are otherwise meant in the complex sense, as the principal root: a^b=Exp[b*Log[a]]. When the base a is negative, the Log is nonreal. Compare

Plot[x^(1/3), {x, -1, 1}]
Plot[CubeRoot[x], {x, -1, 1}]

I always warn my students that the familiar algebraic rules break down when dealing with powers with a negative base and noninteger exponent.