1. The problem asks us to find the value of $d$ in the equality $$\frac{\sqrt{10} \times \sqrt{42}}{\sqrt{2}} = \sqrt{d}$$. 2. Start by simplifying the numerator. Using the property of square roots, $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$, we can write: $$\sqrt{10} \times \sqrt{42} = \sqrt{10 \times 42} = \sqrt{420}$$. 3. Substitute back into the original expression: $$\frac{\sqrt{420}}{\sqrt{2}}$$. 4. Use the property $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$ to combine inside a single square root: $$\sqrt{\frac{420}{2}}$$. 5. Calculate the division inside the root: $$\frac{420}{2} = 210$$. 6. Thus, we have: $$\sqrt{210} = \sqrt{d}$$ which means: $$d = 210$$. Final answer: $$d = 210$$.