1. **State the problem:** Find $n$ such that $$(\sqrt{m})^7 = p \sqrt{m}, \text{ where } p = m^n.$$ 2. **Rewrite the expressions using exponent rules:** Recall that $\sqrt{m} = m^{\frac{1}{2}}$. So, $$(\sqrt{m})^7 = \left(m^{\frac{1}{2}}\right)^7 = m^{\frac{7}{2}}.$$ 3. **Express the right side:** $$p \sqrt{m} = m^n \times m^{\frac{1}{2}} = m^{n + \frac{1}{2}}.$$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, $$\frac{7}{2} = n + \frac{1}{2}.$$ 5. **Solve for $n$:** $$n = \frac{7}{2} - \frac{1}{2} = \frac{6}{2} = 3.$$ **Final answer:** $$\boxed{3}$$