1. **State the problem:** Solve the equation $$(3x - 2)^{\frac{3}{2}} + 7 = 15$$ for $x$. 2. **Isolate the power term:** Subtract 7 from both sides: $$ (3x - 2)^{\frac{3}{2}} = 15 - 7 $$ $$ (3x - 2)^{\frac{3}{2}} = 8 $$ 3. **Rewrite the equation:** Recall that $a^{\frac{3}{2}} = (a^{\frac{1}{2}})^3 = (\sqrt{a})^3$. 4. **Take the cube root of both sides:** $$ \sqrt{3x - 2} = \sqrt[3]{8} $$ $$ \sqrt{3x - 2} = 2 $$ 5. **Square both sides to remove the square root:** $$ (\sqrt{3x - 2})^2 = 2^2 $$ $$ 3x - 2 = 4 $$ 6. **Solve for $x$:** $$ 3x = 4 + 2 $$ $$ 3x = 6 $$ $$ x = \frac{6}{3} $$ $$ x = 2 $$ 7. **Check for extraneous solutions:** Substitute $x=2$ back into the original equation: $$ (3(2) - 2)^{\frac{3}{2}} + 7 = (6 - 2)^{\frac{3}{2}} + 7 = 4^{\frac{3}{2}} + 7 $$ $$ 4^{\frac{3}{2}} = (\sqrt{4})^3 = 2^3 = 8 $$ $$ 8 + 7 = 15 $$ which is true. **Final answer:** $x = 2$