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 \( y=2 \).

Simplify the expression by following steps: - step0: Solution: \(\frac{3ab^{2}}{8xy}\times \frac{16xy^{2}}{21ab^{3}}\) - step1: Reduce the fraction: \(1\times \frac{2y}{7b}\) - step2: Multiply the terms: \(\frac{2y}{7b}\) Calculate or simplify the expression \( \frac{3 x^{2}+6 x}{2 x y+4 y} \times \frac{2 y}{9 x^{2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{3x^{2}+6x}{2xy+4y}\times \frac{2y}{9x^{2}}\) - step1: Divide the terms: \(\frac{3x}{2y}\times \frac{2y}{9x^{2}}\) - step2: Reduce the fraction: \(1\times \frac{1}{3x}\) - step3: Multiply the terms: \(\frac{1}{3x}\) Calculate or simplify the expression \( \frac{36 m^{2} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{36m^{2}n^{3}}{8m}\div \frac{n^{2}m^{2}}{4mn^{3}}\) - step1: Multiply by the reciprocal: \(\frac{36m^{2}n^{3}}{8m}\times \frac{4mn^{3}}{n^{2}m^{2}}\) - step2: Divide the terms: \(\frac{9mn^{3}}{2}\times \frac{4mn^{3}}{n^{2}m^{2}}\) - step3: Divide the terms: \(\frac{9mn^{3}}{2}\times \frac{4n}{m}\) - step4: Reduce the fraction: \(9n^{3}\times 2n\) - step5: Multiply the numbers: \(18n^{3}\times n\) - step6: Multiply the terms: \(18n^{4}\) Calculate or simplify the expression \( \frac{4 a x+8 x}{3 x-1} \div \frac{a+2}{6 x-2} \). Simplify the expression by following steps: - step0: Solution: \(\frac{4ax+8x}{3x-1}\div \frac{a+2}{6x-2}\) - step1: Multiply by the reciprocal: \(\frac{4ax+8x}{3x-1}\times \frac{6x-2}{a+2}\) - step2: Rewrite the expression: \(\frac{4x\left(a+2\right)}{3x-1}\times \frac{6x-2}{a+2}\) - step3: Reduce the fraction: \(\frac{4x}{3x-1}\times \left(6x-2\right)\) - step4: Rewrite the expression: \(\frac{4x}{3x-1}\times 2\left(3x-1\right)\) - step5: Reduce the fraction: \(4x\times 2\) - step6: Multiply the terms: \(8x\) Substitute \( x=-1,a=2 \) into the expression \( 8x \). Evaluate the expression by following steps: - step0: Evaluate: \(8x\) - step1: Substitute: \(8\left(-1\right)\) - step2: Simplify: \(-8\) Substitute \( m=3,n=\frac{1}{3} \) into the expression \( 18n^{4} \). Evaluate the expression by following steps: - step0: Evaluate: \(18n^{4}\) - step1: Substitute: \(18\left(\frac{1}{3}\right)^{4}\) - step2: Evaluate the power: \(18\times \frac{1}{81}\) - step3: Multiply: \(\frac{2}{9}\) Substitute \( x=1,y=2 \) into the expression \( \frac{1}{3x} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{1}{3x}\) - step1: Substitute: \(\frac{1}{3\times 1}\) - step2: Multiply: \(\frac{1}{3}\) Substitute \( b=1,y=2 \) into the expression \( \frac{2y}{7b} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{2y}{7b}\) - step1: Substitute: \(\frac{2\times 2}{7\times 1}\) - step2: Multiply the numbers: \(\frac{4}{7\times 1}\) - step3: Multiply: \(\frac{4}{7}\) Let's solve each of the problems step by step. ### Problem 5 We need to simplify the expression: \[ \frac{3 a b^{2}}{8 x y} \times \frac{16 x y^{2}}{21 a b^{3}} \] **Step 1: Simplify the expression** \[ = \frac{3 \cdot 16 \cdot a \cdot b^{2} \cdot x \cdot y^{2}}{8 \cdot 21 \cdot a \cdot b^{3} \cdot x \cdot y} \] Cancelling out common terms: \[ = \frac{48 b^{2} y}{168 b^{3}} = \frac{2y}{7b} \] **Step 2: Substitute values \( b=1 \) and \( y=2 \)** \[ \frac{2 \cdot 2}{7 \cdot 1} = \frac{4}{7} \] ### Problem 6 We need to simplify the expression: \[ \frac{4 a x + 8 x}{3 x - 1} \div \frac{a + 2}{6 x - 2} \] **Step 1: Simplify the expression** \[ = \frac{(4 a + 8)x}{3 x - 1} \cdot \frac{6 x - 2}{a + 2} \] Factoring out common terms: \[ = \frac{(4a + 8)x(6x - 2)}{(3x - 1)(a + 2)} \] This simplifies to: \[ = 8x \] **Step 2: Substitute values \( x=-1 \) and \( a=2 \)** \[ 8 \cdot (-1) = -8 \] ### Problem 7 We need to simplify the expression: \[ \frac{36 m^{2} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} \] **Step 1: Simplify the expression** \[ = \frac{36 m^{2} n^{3}}{8 m} \cdot \frac{4 m n^{3}}{n^{2} m^{2}} = \frac{36 \cdot 4 n^{3}}{8 n^{2}} = \frac{144 n^{3}}{8 n^{2}} = 18 n^{4} \] **Step 2: Substitute values \( m=3 \) and \( n=\frac{1}{3} \)** \[ 18 \left(\frac{1}{3}\right)^{4} = 18 \cdot \frac{1}{81} = \frac{2}{9} \] ### Problem 8 We need to simplify the expression: \[ \frac{3 x^{2} + 6 x}{2 x y + 4 y} \times \frac{2 y}{9 x^{2}} \] **Step 1: Simplify the expression** \[ = \frac{3x(x + 2)}{2y(x + 2)} \cdot \frac{2y}{9x^{2}} = \frac{3x \cdot 2y}{2y \cdot 9x^{2}} = \frac{1}{3x} \] **Step 2: Substitute values \( x=1 \) and \( y=2 \)** \[ \frac{1}{3 \cdot 1} = \frac{1}{3} \] ### Summary of Results 1. Problem 5: \(\frac{4}{7}\) 2. Problem 6: \(-8\) 3. Problem 7: \(\frac{2}{9}\) 4. Problem 8: \(\frac{1}{3}\)