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We do not know what is sol, but i suppose it is the following...

sage: var('r');
sage: eq = 3*(2.2 + (64/r)^(1/3)) == 4*(2.2 + (128/(r-1))^(1/4))
sage: sol = solve(eq, r)
sage: sol
[r^(1/3) == 60*(r - 1)^(1/4)/(40*8^(1/4) + 11*(r - 1)^(1/4))]


It is clear that we cannot ask for the numerical value of sol[0].

Instead, we could help sage deliver the needed value. We just substitute $$r = R^3$$ and rewrite the given equation in the form: $$# \frac 34\left(\frac{22}{10} +\frac 4R\right)-\frac {22}{10} \left(\frac {128}{R^3-1}\right)^{1/4}\ .$$ Rising both sides to the fourth power leads to a polynomial equation, which we can solve. We obtain one value of $R$, introduced by taking the fourth power. But it leads to no solution in $r$.

Try a plot of the difference to see there is no solution.

If there is some typo, then fix it, find an interval where the difference function changes the sign, then use gp-solve. (Inside pari or inside sage.)

We do not know what is sol, but i suppose it is the following...

sage: var('r');
sage: eq = 3*(2.2 + (64/r)^(1/3)) == 4*(2.2 + (128/(r-1))^(1/4))
sage: sol = solve(eq, r)
sage: sol
[r^(1/3) == 60*(r - 1)^(1/4)/(40*8^(1/4) + 11*(r - 1)^(1/4))]


It is clear that we cannot ask for the numerical value of sol[0].

Instead, we could help sage deliver the needed value. We just substitute $$r = R^3$$ and rewrite the given equation in the form: $$#$$ \frac 34\left(\frac{22}{10} +\frac 4R\right)-\frac {22}{10}

{22}{10} = \left(\frac {128}{R^3-1}\right)^{1/4}\ .  Rising both sides to the fourth power leads to a polynomial equation, which we can solve. We obtain one value of $R$, introduced by taking the fourth power. But it leads to no solution in $r$.

Try a plot of the difference to see there is no solution.

If there is some typo, then fix it, find an interval where the difference function changes the sign, then use gp-solve. (Inside pari or inside sage.)

sage.)