Question:

Step 1: Determine the volume of the regular tetrahedron. The volume $V$ of a regular tetrahedron with edge length $a$ is given by the formula: $$V = \frac{a^3\sqrt{2}}{12}$$ Given that the edge length $a=1$, substitute this value into the formula: $$V = \frac{(1)^3\sqrt{2}}{12} = \frac{\sqrt{2}}{12}$$ Step 2: Calculate $6V$. Multiply the volume by 6: $$6V = 6 \cdot \frac{\sqrt{2}}{12} = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2}$$ Step 3: Express the result in the form $\frac{p}{q}$. The expression $6V = \frac{\sqrt{2}}{2}$ contains an irrational number, $\sqrt{2}$, and thus cannot be directly written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. The problem statement implies that a rational form is expected. This suggests that the quantity to be expressed as $\frac{p}{q}$ is $(6V)^2$. Calculate $(6V)^2$: $$(6V)^2 = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{(\sqrt{2})^2}{2^2} = \frac{2}{4} = \frac{1}{2}$$ Now, we have $(6V)^2 = \frac{1}{2}$. Comparing this to $\frac{p}{q}$, we identify $p=1$ and $q=2$. These values $p=1$ and $q=2$ are relatively prime. Step 4: Calculate $(p+q)$. Using the identified values $p=1$ and $q=2$: $$p+q = 1+2 = 3$$