1. **State the problem:** We have a multiplication grid with unknowns $a$, $b$, $c$, and $d$. The grid shows products of numbers in the top row and left column. 2. **Understand the grid:** The top row is $2$, $a$, $b$ and the left column is $\sqrt{7}$, $3\sqrt{5}$. The products are given as: - $2 \times \sqrt{7} = 2\sqrt{7}$ - $a \times \sqrt{7} = c$ - $b \times \sqrt{7} = 5\sqrt{42}$ - $2 \times 3\sqrt{5} = 6\sqrt{5}$ - $a \times 3\sqrt{5} = 3\sqrt{15}$ - $b \times 3\sqrt{5} = d$ 3. **Find $a$:** From $a \times 3\sqrt{5} = 3\sqrt{15}$, divide both sides by $3\sqrt{5}$: $$a = \frac{3\sqrt{15}}{3\sqrt{5}} = \frac{\sqrt{15}}{\sqrt{5}} = \sqrt{\frac{15}{5}} = \sqrt{3}$$ 4. **Find $c$:** From $a \times \sqrt{7} = c$ and $a = \sqrt{3}$: $$c = \sqrt{3} \times \sqrt{7} = \sqrt{21}$$ 5. **Find $b$:** From $b \times \sqrt{7} = 5\sqrt{42}$, divide both sides by $\sqrt{7}$: $$b = \frac{5\sqrt{42}}{\sqrt{7}} = 5 \sqrt{\frac{42}{7}} = 5\sqrt{6}$$ 6. **Find $d$:** From $b \times 3\sqrt{5} = d$ and $b = 5\sqrt{6}$: $$d = 5\sqrt{6} \times 3\sqrt{5} = 15 \sqrt{30}$$ **Final answers:** - $a = \sqrt{3}$ - $b = 5\sqrt{6}$ - $c = \sqrt{21}$ - $d = 15\sqrt{30}$