Question: Simplify.
$\frac{\sqrt{-10} \cdot \sqrt{-5}}{\sqrt{2}}$
1. **State the problem:** Simplify the expression $$\frac{\sqrt{-10} \cdot \sqrt{-5}}{\sqrt{2}}$$.
2. **Recall the property of square roots:** For any numbers $a$ and $b$, $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$, even if $a$ and $b$ are negative, but we must consider imaginary units for negative values.
3. **Multiply the square roots in the numerator:**
$$\sqrt{-10} \cdot \sqrt{-5} = \sqrt{(-10) \cdot (-5)} = \sqrt{50}$$
4. **Rewrite the expression:**
$$\frac{\sqrt{50}}{\sqrt{2}}$$
5. **Use the property of division of square roots:**
$$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}}$$
6. **Simplify the fraction inside the square root:**
$$\sqrt{\frac{50}{2}} = \sqrt{25}$$
7. **Calculate the square root:**
$$\sqrt{25} = 5$$
8. **Final answer:**
$$5$$
**Note:** The imaginary units cancel out because $\sqrt{-10} = \sqrt{10}i$ and $\sqrt{-5} = \sqrt{5}i$, so their product is $\sqrt{10} \cdot \sqrt{5} \cdot i^2 = \sqrt{50} \cdot (-1) = -\sqrt{50}$, but since the problem uses the property directly, the simplification leads to $5i^2 = -5$. To be precise, let's include the imaginary unit step:
- $\sqrt{-10} = \sqrt{10}i$
- $\sqrt{-5} = \sqrt{5}i$
- So, $\sqrt{-10} \cdot \sqrt{-5} = \sqrt{10}i \cdot \sqrt{5}i = \sqrt{50} i^2 = \sqrt{50} \cdot (-1) = -\sqrt{50}$
Then the expression is:
$$\frac{-\sqrt{50}}{\sqrt{2}} = -\sqrt{\frac{50}{2}} = -\sqrt{25} = -5$$
So the correct final answer is:
$$-5$$