1. **State the problem:** Given that $3 \sqrt{7} = \sqrt{m}$, find the value of $m$. 2. **Recall the property of square roots:** For any positive numbers $a$ and $b$, $a \sqrt{b} = \sqrt{a^2 b}$. 3. **Apply the property:** $$3 \sqrt{7} = \sqrt{3^2 \times 7} = \sqrt{9 \times 7} = \sqrt{63}$$ 4. **Equate the expressions:** Since $3 \sqrt{7} = \sqrt{m}$, it follows that $$\sqrt{m} = \sqrt{63}$$ 5. **Conclude the value of $m$:** $$m = 63$$ **Final answer:** $m = 63$