1. **State the problem:** We need to make $x$ the subject of the formula given by $$2y^2 = \frac{5(x - 3)^5}{3y}.$$ 2. **Write down the formula and explain:** The goal is to isolate $x$ on one side. The formula involves powers and fractions, so we will use algebraic operations like multiplication, division, and taking roots. 3. **Multiply both sides by $3y$ to eliminate the denominator:** $$3y \times 2y^2 = 3y \times \frac{5(x - 3)^5}{3y}$$ $$6y^3 = 5(x - 3)^5$$ 4. **Divide both sides by 5 to isolate the power term:** $$\frac{6y^3}{5} = \frac{5(x - 3)^5}{\cancel{5}}$$ $$\frac{6y^3}{5} = (x - 3)^5$$ 5. **Take the fifth root of both sides to solve for $(x - 3)$:** $$\sqrt[5]{\frac{6y^3}{5}} = x - 3$$ 6. **Add 3 to both sides to isolate $x$:** $$x = 3 + \sqrt[5]{\frac{6y^3}{5}}$$ **Final answer:** $$\boxed{x = 3 + \sqrt[5]{\frac{6y^3}{5}}}$$