Question: Simplify the radical. Assume that all variables represent positive real numbers.
$\sqrt{49a^4b^{40}}$
1. **State the problem:** Simplify the expression $$\sqrt{49a^4b^{40}}$$ assuming all variables are positive real numbers.
2. **Recall the formula:** For any positive real number $x$, $$\sqrt{x^2} = x$$ and $$\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}$$.
3. **Apply the square root to each factor:**
$$\sqrt{49a^4b^{40}} = \sqrt{49} \cdot \sqrt{a^4} \cdot \sqrt{b^{40}}$$
4. **Simplify each square root:**
- $$\sqrt{49} = 7$$ because $7^2 = 49$
- $$\sqrt{a^4} = a^{\frac{4}{2}} = a^2$$
- $$\sqrt{b^{40}} = b^{\frac{40}{2}} = b^{20}$$
5. **Combine the simplified terms:**
$$7 \cdot a^2 \cdot b^{20} = 7a^2b^{20}$$
6. **Final answer:**
$$\sqrt{49a^4b^{40}} = 7a^2b^{20}$$