1. State the problem: Compute $1\sqrt{126}$.\n2. Formula and rules: Use the product rule for square roots $\sqrt{ab}=\sqrt{a}\sqrt{b}$.\n3. Factor 126 into primes: $126=2\cdot3^2\cdot7$.\n4. Extract perfect squares: $\sqrt{126}=\sqrt{2\cdot3^2\cdot7}=\sqrt{3^2}\sqrt{2\cdot7}=3\sqrt{14}$.\n5. Multiply by 1 and simplify: $1\sqrt{126}=1\cdot\sqrt{126}=\sqrt{126}=3\sqrt{14}$.\n6. Final answer: $3\sqrt{14}$.\n