1. The problem asks to write the root expression $7\sqrt{7}$ in index (exponent) form. 2. Recall that the square root of a number $a$ can be written as $a^{\frac{1}{2}}$. 3. Therefore, $\sqrt{7} = 7^{\frac{1}{2}}$. 4. The expression $7\sqrt{7}$ can be rewritten as $7 \times 7^{\frac{1}{2}}$. 5. Using the rule of exponents $a^m \times a^n = a^{m+n}$, we add the exponents: $7^{1} \times 7^{\frac{1}{2}} = 7^{1 + \frac{1}{2}}$. 6. Simplify the exponent: $1 + \frac{1}{2} = \frac{3}{2}$. 7. Thus, $7\sqrt{7} = 7^{\frac{3}{2}}$. Final answer: $$7\sqrt{7} = 7^{\frac{3}{2}}$$