1. **State the problem:** Simplify the expression $$\sqrt[4]{16 a^4 b^{16}}$$ where $a$ and $b$ are real numbers. 2. **Recall the formula:** The fourth root of a product is the product of the fourth roots: $$\sqrt[4]{x y} = \sqrt[4]{x} \cdot \sqrt[4]{y}$$. 3. **Apply the fourth root to each factor:** $$\sqrt[4]{16} \cdot \sqrt[4]{a^4} \cdot \sqrt[4]{b^{16}}$$ 4. **Simplify each term:** - $16 = 2^4$, so $$\sqrt[4]{16} = 2$$. - $$\sqrt[4]{a^4} = |a|$$ because the fourth root of $a^4$ is $|a|$ to ensure the result is nonnegative. - $$\sqrt[4]{b^{16}} = |b|^4$$ since $$\sqrt[4]{b^{16}} = b^{16/4} = b^4$$ and for even roots, absolute value is used: $$|b|^4$$. 5. **Combine the simplified terms:** $$2 \cdot |a| \cdot |b|^4$$ 6. **Final answer:** $$2 |a| |b|^4$$