5. Write \( \frac{3 a b^{2}}{8 x y} \times \frac{16 x y^{2}}{21 a b^{3}} \) in its simplified forn and substitute the values to evaluate, given that \( b=1 \) and \( y=2 \). 6. Write \( \frac{4 a x+8 x}{3 x-1} \div \frac{a+2}{6 x-2} \) in its simplified form and substitute the value to evaluate, given that \( x=-1 \) and \( a=2 \). 7. Write \( \frac{36 m^{2} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} \) in its simplest form and substitute the values to evaluate, given that \( m=3 \) and \( n=\frac{1}{3} \) 8. Write \( \frac{3 x^{2}+6 x}{2 x y+4 y} \times \frac{2 y}{9 x^{2}} \) in its simplified form and substitute the value to evaluate, given that \( x=1 \) and \( y=2 \).

To simplify \( \frac{3 a b^{2}}{8 x y} \times \frac{16 x y^{2}}{21 a b^{3}} \), we can cancel common factors. The expression simplifies to \( \frac{3 \cdot 16 \cdot b^{2} \cdot y^{2}}{8 \cdot 21 \cdot b^{3}} \), or \( \frac{48 y^{2}}{168 b} \). This reduces to \( \frac{y^{2}}{3.5 b} \). Substituting \( b=1 \) and \( y=2 \), we have \( \frac{2^{2}}{3.5 \cdot 1} = \frac{4}{3.5} = \frac{8}{7} \). For the expression \( \frac{4 a x+8 x}{3 x-1} \div \frac{a+2}{6 x-2} \), we first rewrite it as \( \frac{(4 a x + 8 x)(6 x - 2)}{(3 x - 1)(a + 2)} \). Factoring the numerator gives us \( \frac{4x(6(a + 2) - 2)}{(3x - 1)(a + 2)} \). We can now substitute \( a=2 \) and \( x=-1 \) to simplify and evaluate the final expression efficiently. For the simplification of \( \frac{36 m^{2} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} \), we rewrite it as \( \frac{36 m^{2} n^{3}}{8 m} \times \frac{4 m n^{3}}{n^{2} m^{2}} \). This reduces to \( \frac{36 \cdot 4 n^{5}}{8 n^{2}} = \frac{144 n^{5}}{8 n^{2}} \), simplifying down to \( 18 n^{3} \). Substituting \( m=3 \) and \( n=\frac{1}{3} \), we calculate \( 18 \left(\frac{1}{3}\right)^{3} = 18 \cdot \frac{1}{27} = \frac{2}{3} \). For the last expression \( \frac{3 x^{2}+6 x}{2 x y+4 y} \times \frac{2 y}{9 x^{2}} \), we factor the numerator and denominator, resulting in \( \frac{3(x + 2)}{2y(x + 2)} \times \frac{2y}{9x^{2}} \), which simplifies to \( \frac{y}{3x^{2}} \). By substituting \( x=1 \) and \( y=2 \), the expression evaluates to \( \frac{2}{3 \cdot 1^{2}} = \frac{2}{3} \).