1. **State the problem:** Solve the equation $$x^{\frac{7}{3}} = 128$$ for $x$. 2. **Recall the formula:** To solve for $x$ when it is raised to a fractional power, use the property: $$x^{\frac{m}{n}} = a \implies x = a^{\frac{n}{m}}$$ where $m$ and $n$ are integers. 3. **Apply the formula:** Here, $m=7$ and $n=3$, and $a=128$. So, $$x = 128^{\frac{3}{7}}$$ 4. **Simplify the base:** Note that $128 = 2^7$. 5. **Substitute and simplify:** $$x = (2^7)^{\frac{3}{7}} = 2^{7 \times \frac{3}{7}} = 2^3$$ 6. **Calculate the final value:** $$2^3 = 8$$ **Final answer:** $$x = 8$$