1. **State the problem:** Find the product of the given expressions: - $\left(\sqrt{x^2 - y^2}\right)^7 \sqrt{(16x^2 - 16y^2)^2}$ - $(-2 - 5\sqrt{2})(-2 + 5\sqrt{2})$ - $\left(5 \sqrt[3]{m^2} - \sqrt[3]{4n}\right)\left(25m^3\sqrt{m} + 5 \sqrt[3]{4m^2n} + \sqrt[3]{16n^2}\right)$ 2. **Simplify the first expression:** - Note that $\sqrt{(16x^2 - 16y^2)^2} = |16x^2 - 16y^2| = 16|x^2 - y^2|$ - Also, $\left(\sqrt{x^2 - y^2}\right)^7 = (x^2 - y^2)^{7/2}$ - So the product is: $$ (x^2 - y^2)^{7/2} \times 16|x^2 - y^2| = 16 (x^2 - y^2)^{7/2} |x^2 - y^2| $$ - Since $|x^2 - y^2| = (x^2 - y^2)$ if $x^2 - y^2 \geq 0$, the expression becomes: $$ 16 (x^2 - y^2)^{7/2 + 1} = 16 (x^2 - y^2)^{9/2} $$ 3. **Simplify the second expression:** - Use the difference of squares formula: $$ (a - b)(a + b) = a^2 - b^2 $$ - Here, $a = -2$, $b = 5\sqrt{2}$ - Calculate: $$ (-2)^2 - (5\sqrt{2})^2 = 4 - 25 \times 2 = 4 - 50 = -46 $$ 4. **Simplify the third expression:** - Let $a = 5 \sqrt[3]{m^2}$ and $b = \sqrt[3]{4n}$ - The product is: $$ (a - b)(a^2 + ab + b^2) = a^3 - b^3 $$ - Calculate $a^3$: $$ (5 \sqrt[3]{m^2})^3 = 5^3 (m^2) = 125 m^2 $$ - Calculate $b^3$: $$ (\sqrt[3]{4n})^3 = 4n $$ - So the product is: $$ 125 m^2 - 4 n $$ **Final answers:** - First expression: $16 (x^2 - y^2)^{9/2}$ - Second expression: $-46$ - Third expression: $125 m^2 - 4 n$ These are the simplified products of the given expressions.