1. The problem asks to find the exact value of $\tan 30^\circ \times \tan 30^\circ$ in simplest form. 2. Recall the exact value of $\tan 30^\circ$ is $\frac{1}{\sqrt{3}}$. 3. Using this, we calculate: $$\tan 30^\circ \times \tan 30^\circ = \left(\frac{1}{\sqrt{3}}\right) \times \left(\frac{1}{\sqrt{3}}\right)$$ 4. Multiply the fractions: $$= \frac{1 \times 1}{\sqrt{3} \times \sqrt{3}} = \frac{1}{3}$$ 5. The value $\frac{1}{3}$ is already in simplest form. Therefore, the exact value of $\tan 30^\circ \times \tan 30^\circ$ is $\frac{1}{3}$.