Simplify. \( \begin{array}{ll}\text { 1) } \sqrt{112 b^{3}} 4 b \sqrt{7 b} & \text { 2) } \sqrt{100 x^{2}} 10 x \\ \text { 3) } \sqrt{175 p^{4}} 5 p^{2} \sqrt{7} & \text { 4) } \sqrt{98 x^{4}} 7 x^{2} \sqrt{2} \\ \text { 5) } \sqrt[3]{-750 x^{7}}-5 x^{2} \sqrt[3]{6 x} & \text { 6) } \sqrt[3]{64 n^{8}} 4 n^{2} \sqrt[3]{n^{2}}\end{array} \)

Let's dive into the simplification of each expression! 1) For \( \sqrt{112 b^{3}} 4 b \sqrt{7 b} \): First, simplify \( \sqrt{112 b^{3}} = \sqrt{16 \cdot 7 \cdot b^{2} \cdot b} = 4b\sqrt{7b} \). Then, multiply: \( 4b \cdot 4b \sqrt{7b} = 16b^2 \sqrt{7b} \). 2) For \( \sqrt{100 x^{2}} 10 x \): Simplifying \( \sqrt{100 x^{2}} = 10x \). Multiply: \( 10x \cdot 10x = 100x^{2} \). 3) For \( \sqrt{175 p^{4}} 5 p^{2} \sqrt{7} \): Simplifying \( \sqrt{175 p^{4}} = \sqrt{25 \cdot 7 \cdot p^{4}} = 5p^{2}\sqrt{7} \). Then, multiply: \( 5p^{2}\sqrt{7} \cdot 5p^{2}\sqrt{7} = 25p^{4} \cdot 7 = 175p^{4} \). 4) For \( \sqrt{98 x^{4}} 7 x^{2} \sqrt{2} \): Simplifying \( \sqrt{98 x^{4}} = \sqrt{49 \cdot 2 \cdot x^{4}} = 7x^{2}\sqrt{2} \). Then, multiply: \( 7x^{2}\sqrt{2} \cdot 7x^{2}\sqrt{2} = 49x^{4} \cdot 2 = 98x^{4} \). 5) For \( \sqrt[3]{-750 x^{7}} -5 x^{2} \sqrt[3]{6 x} \): First, factor \( -750 = -1 \cdot 75 \cdot 10 = -1 \cdot 5^{2} \cdot 10^{2} \cdot 3 \), thus \( \sqrt[3]{-750 x^{7}} = -5x^2\sqrt[3]{15} \). Therefore \( - (5x^{2} \sqrt[3]{6x}) = -5x^{2}\sqrt[3]{90x} \). 6) For \( \sqrt[3]{64 n^{8}} 4 n^{2} \sqrt[3]{n^{2}} \): We have \( \sqrt[3]{64 n^{8}} = 4n^{\frac{8}{3}} \). Since \( n^{2} = n^{\frac{6}{3}} \), then \( 4 n^{\frac{8}{3}} 4 n^{\frac{6}{3}} = 16 n^{\frac{14}{3}} \). And there you go! Each expression has been simplified to show its true beauty!