1. **State the problem:** Solve the equation $$(x - 3)^{\frac{3}{5}} = 8$$ for $x$. 2. **Isolate the expression:** The equation is already isolated with the power expression on one side. 3. **Eliminate the fractional exponent:** To remove the fractional exponent $\frac{3}{5}$, raise both sides of the equation to the reciprocal power $\frac{5}{3}$: $$\left((x - 3)^{\frac{3}{5}}\right)^{\frac{5}{3}} = 8^{\frac{5}{3}}$$ 4. **Simplify the left side:** Using the property $(a^m)^n = a^{mn}$, $$ (x - 3)^{\frac{3}{5} \times \frac{5}{3}} = (x - 3)^1 = x - 3 $$ 5. **Calculate the right side:** $$ 8^{\frac{5}{3}} = \left(8^{\frac{1}{3}}\right)^5 $$ Since $8^{\frac{1}{3}} = 2$ (cube root of 8), $$ 2^5 = 32 $$ 6. **Set the simplified equation:** $$ x - 3 = 32 $$ 7. **Solve for $x$:** $$ x = 32 + 3 = 35 $$ **Final answer:** $x = 35$