1. **State the problem:** Find the positive solution of the equation $$7x^{\frac{5}{3}} + 27 = 54459$$. 2. **Isolate the term with the variable:** Subtract 27 from both sides: $$7x^{\frac{5}{3}} + 27 - 27 = 54459 - 27$$ $$7x^{\frac{5}{3}} = 54432$$ 3. **Divide both sides by 7 to solve for $x^{\frac{5}{3}}$:** $$\cancel{7}x^{\frac{5}{3}} = \frac{54432}{\cancel{7}}$$ $$x^{\frac{5}{3}} = 7776$$ 4. **Rewrite the equation to solve for $x$:** Recall that $x^{\frac{5}{3}} = (x^{\frac{1}{3}})^5$, so $$x^{\frac{1}{3}} = \sqrt[5]{7776}$$ 5. **Calculate $\sqrt[5]{7776}$:** Prime factorize 7776: $$7776 = 6^5$$ So, $$\sqrt[5]{7776} = \sqrt[5]{6^5} = 6$$ 6. **Now solve for $x$:** Since $x^{\frac{1}{3}} = 6$, cube both sides: $$\left(x^{\frac{1}{3}}\right)^3 = 6^3$$ $$x = 216$$ **Final answer:** $$\boxed{216}$$