Question: Simplify the following radical expression.
$14\sqrt{23} \cdot 5\sqrt{2}$
1. **State the problem:** Simplify the expression $14\sqrt{23} \cdot 5\sqrt{2}$.
2. **Recall the multiplication rule for radicals:** For any positive numbers $a$, $b$, $c$, and $d$, we have:
$$a\sqrt{b} \cdot c\sqrt{d} = (a \cdot c) \sqrt{b \cdot d}$$
3. **Apply the rule:**
$$14\sqrt{23} \cdot 5\sqrt{2} = (14 \cdot 5) \sqrt{23 \cdot 2}$$
4. **Calculate the products:**
$$14 \cdot 5 = 70$$
$$23 \cdot 2 = 46$$
5. **Substitute back:**
$$70 \sqrt{46}$$
6. **Check if the radical can be simplified:**
$46 = 2 \times 23$, both prime factors, so $\sqrt{46}$ cannot be simplified further.
**Final answer:**
$$70 \sqrt{46}$$