1. **Problem:** Simplify $\left(2^{\frac{1}{3}}\right)^7$. 2. **Formula:** When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$. 3. **Step:** Here, we have seven factors of $2^{\frac{1}{3}}$, so the exponent is $7 \times \frac{1}{3} = \frac{7}{3}$. 4. **Simplification:** $$\left(2^{\frac{1}{3}}\right)^7 = 2^{7 \times \frac{1}{3}} = 2^{\frac{7}{3}}$$ 5. **Interpretation:** This means the cube root of 2 raised to the 7th power, or equivalently, $2^2 \times 2^{\frac{1}{3}} = 4 \times 2^{\frac{1}{3}}$. 6. **Final answer:** $$2^{\frac{7}{3}}$$