EXERCISE (a) Simplify the followiag \( \begin{array}{lll}\text { (1) } \frac{2 x}{3}+\frac{1}{6} & \text { (2) } \frac{7}{6 x}-\frac{2}{9 y}+\frac{1}{3 x^{2}} & \text { (3) } \frac{3}{x}+\frac{2}{x^{2}} \\ \text { (4) } 1+2-\frac{2 a-1}{2} & \text { (5) } \frac{5}{6 x y}-\frac{x+2}{2 x} & \text { (6) } \frac{5}{2 x}-\frac{x+1}{x-2} \\ \text { (7) } \frac{2}{x}+\frac{5}{x+1} & \text { (8) } \frac{x+2}{x^{2}+2}-\frac{6}{x+2} & \text { (9) } \frac{x-3}{x+3}-\frac{x+3}{x-3} \\ \text { (10) } \frac{4}{(2 x+1)^{2}}-\frac{x+1}{2 x+1} & \text { (11) } \frac{2}{2 x}+\frac{7 x+1}{4 x^{2} y}-\frac{5 x y}{4}+1 \\ \text { (12) } \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} & \text { (13) } x-\frac{2 x}{3 x-2}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{3}{x}+\frac{2}{x^{2}}\) - step1: Reduce fractions to a common denominator: \(\frac{3x}{x\times x}+\frac{2}{x^{2}}\) - step2: Multiply the terms: \(\frac{3x}{x^{2}}+\frac{2}{x^{2}}\) - step3: Transform the expression: \(\frac{3x+2}{x^{2}}\) Calculate or simplify the expression \( 5/(2*x) - (x + 1)/(x - 2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{5}{2x}-\frac{\left(x+1\right)}{\left(x-2\right)}\) - step1: Remove the parentheses: \(\frac{5}{2x}-\frac{x+1}{x-2}\) - step2: Reduce fractions to a common denominator: \(\frac{5\left(x-2\right)}{2x\left(x-2\right)}-\frac{\left(x+1\right)\times 2x}{\left(x-2\right)\times 2x}\) - step3: Reorder the terms: \(\frac{5\left(x-2\right)}{2x\left(x-2\right)}-\frac{\left(x+1\right)\times 2x}{2\left(x-2\right)x}\) - step4: Rewrite the expression: \(\frac{5\left(x-2\right)}{2x\left(x-2\right)}-\frac{\left(x+1\right)\times 2x}{2x\left(x-2\right)}\) - step5: Transform the expression: \(\frac{5\left(x-2\right)-\left(x+1\right)\times 2x}{2x\left(x-2\right)}\) - step6: Multiply the terms: \(\frac{5x-10-\left(x+1\right)\times 2x}{2x\left(x-2\right)}\) - step7: Multiply the terms: \(\frac{5x-10-\left(2x^{2}+2x\right)}{2x\left(x-2\right)}\) - step8: Subtract the terms: \(\frac{3x-10-2x^{2}}{2x\left(x-2\right)}\) - step9: Multiply the terms: \(\frac{3x-10-2x^{2}}{2x^{2}-4x}\) Calculate or simplify the expression \( x - (2*x)/(3*x - 2) \). Simplify the expression by following steps: - step0: Solution: \(x-\frac{2x}{\left(3x-2\right)}\) - step1: Remove the parentheses: \(x-\frac{2x}{3x-2}\) - step2: Reduce fractions to a common denominator: \(\frac{x\left(3x-2\right)}{3x-2}-\frac{2x}{3x-2}\) - step3: Transform the expression: \(\frac{x\left(3x-2\right)-2x}{3x-2}\) - step4: Multiply the terms: \(\frac{3x^{2}-2x-2x}{3x-2}\) - step5: Subtract the terms: \(\frac{3x^{2}-4x}{3x-2}\) Calculate or simplify the expression \( 2/x + 5/(x + 1) \). Simplify the expression by following steps: - step0: Solution: \(\frac{2}{x}+\frac{5}{\left(x+1\right)}\) - step1: Remove the parentheses: \(\frac{2}{x}+\frac{5}{x+1}\) - step2: Reduce fractions to a common denominator: \(\frac{2\left(x+1\right)}{x\left(x+1\right)}+\frac{5x}{\left(x+1\right)x}\) - step3: Rewrite the expression: \(\frac{2\left(x+1\right)}{x\left(x+1\right)}+\frac{5x}{x\left(x+1\right)}\) - step4: Transform the expression: \(\frac{2\left(x+1\right)+5x}{x\left(x+1\right)}\) - step5: Multiply the terms: \(\frac{2x+2+5x}{x\left(x+1\right)}\) - step6: Add the terms: \(\frac{7x+2}{x\left(x+1\right)}\) - step7: Multiply the terms: \(\frac{7x+2}{x^{2}+x}\) Calculate or simplify the expression \( (x - 3)/(x + 3) - (x + 3)/(x - 3) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-3\right)}{\left(x+3\right)}-\frac{\left(x+3\right)}{\left(x-3\right)}\) - step1: Remove the parentheses: \(\frac{x-3}{x+3}-\frac{x+3}{x-3}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(x-3\right)\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}-\frac{\left(x+3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}\) - step3: Rewrite the expression: \(\frac{\left(x-3\right)\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}-\frac{\left(x+3\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}\) - step4: Transform the expression: \(\frac{\left(x-3\right)\left(x-3\right)-\left(x+3\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}\) - step5: Multiply the terms: \(\frac{x^{2}-6x+9-\left(x+3\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}\) - step6: Multiply the terms: \(\frac{x^{2}-6x+9-\left(x^{2}+6x+9\right)}{\left(x+3\right)\left(x-3\right)}\) - step7: Calculate: \(\frac{-12x}{\left(x+3\right)\left(x-3\right)}\) - step8: Rewrite the fraction: \(-\frac{12x}{\left(x+3\right)\left(x-3\right)}\) - step9: Multiply the terms: \(-\frac{12x}{x^{2}-9}\) Calculate or simplify the expression \( (x + 2)/(x^2 + 2) - 6/(x + 2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+2\right)}{\left(x^{2}+2\right)}-\frac{6}{\left(x+2\right)}\) - step1: Remove the parentheses: \(\frac{x+2}{x^{2}+2}-\frac{6}{x+2}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(x+2\right)\left(x+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}-\frac{6\left(x^{2}+2\right)}{\left(x+2\right)\left(x^{2}+2\right)}\) - step3: Rewrite the expression: \(\frac{\left(x+2\right)\left(x+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}-\frac{6\left(x^{2}+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step4: Transform the expression: \(\frac{\left(x+2\right)\left(x+2\right)-6\left(x^{2}+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step5: Multiply the terms: \(\frac{x^{2}+4x+4-6\left(x^{2}+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step6: Multiply the terms: \(\frac{x^{2}+4x+4-\left(6x^{2}+12\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step7: Calculate: \(\frac{-5x^{2}+4x-8}{\left(x^{2}+2\right)\left(x+2\right)}\) - step8: Multiply the terms: \(\frac{-5x^{2}+4x-8}{x^{3}+2x^{2}+2x+4}\) Calculate or simplify the expression \( 2/(2*x) + (7*x + 1)/(4*x^2*y) - (5*x*y)/4 + 1 \). Simplify the expression by following steps: - step0: Solution: \(\frac{2}{2x}+\frac{\left(7x+1\right)}{4x^{2}y}-\frac{5xy}{4}+1\) - step1: Remove the parentheses: \(\frac{2}{2x}+\frac{7x+1}{4x^{2}y}-\frac{5xy}{4}+1\) - step2: Calculate: \(\frac{1}{x}+\frac{7x+1}{4x^{2}y}-\frac{5xy}{4}+1\) - step3: Reduce fractions to a common denominator: \(\frac{4xy}{x\times 4xy}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4xyx}+\frac{4xyx}{4xyx}\) - step4: Reorder the terms: \(\frac{4xy}{4x\times xy}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4xyx}+\frac{4xyx}{4xyx}\) - step5: Multiply the terms: \(\frac{4xy}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4xyx}+\frac{4xyx}{4xyx}\) - step6: Multiply the terms: \(\frac{4xy}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4x^{2}y}+\frac{4xyx}{4xyx}\) - step7: Multiply the terms: \(\frac{4xy}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4x^{2}y}+\frac{4xyx}{4x^{2}y}\) - step8: Transform the expression: \(\frac{4xy+7x+1-5xyxyx+4xyx}{4x^{2}y}\) - step9: Multiply the terms: \(\frac{4xy+7x+1-5x^{3}y^{2}+4xyx}{4x^{2}y}\) - step10: Multiply the terms: \(\frac{4xy+7x+1-5x^{3}y^{2}+4x^{2}y}{4x^{2}y}\) - step11: Simplify: \(\frac{4xy+7x+1-5x^{3}y^{2}+4x^{2}y}{4yx^{2}}\) Calculate or simplify the expression \( (2*x)/3 + 1/6 \). Simplify the expression by following steps: - step0: Solution: \(\frac{2x}{3}+\frac{1}{6}\) - step1: Reduce fractions to a common denominator: \(\frac{2x\times 2}{3\times 2}+\frac{1}{6}\) - step2: Multiply the numbers: \(\frac{2x\times 2}{6}+\frac{1}{6}\) - step3: Transform the expression: \(\frac{2x\times 2+1}{6}\) - step4: Multiply the terms: \(\frac{4x+1}{6}\) Calculate or simplify the expression \( 4/((2*x + 1)^2) - (x + 1)/(2*x + 1) \). Simplify the expression by following steps: - step0: Solution: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{\left(x+1\right)}{\left(2x+1\right)}\) - step1: Remove the parentheses: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{x+1}{2x+1}\) - step2: Reduce fractions to a common denominator: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)\left(2x+1\right)}\) - step3: Multiply: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^{2}}\) - step4: Transform the expression: \(\frac{4-\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^{2}}\) - step5: Multiply the terms: \(\frac{4-\left(2x^{2}+3x+1\right)}{\left(2x+1\right)^{2}}\) - step6: Subtract the terms: \(\frac{3-2x^{2}-3x}{\left(2x+1\right)^{2}}\) - step7: Calculate: \(\frac{3-2x^{2}-3x}{4x^{2}+4x+1}\) Calculate or simplify the expression \( 7/(6*x) - 2/(9*y) + 1/(3*x^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{7}{6x}-\frac{2}{9y}+\frac{1}{3x^{2}}\) - step1: Reduce fractions to a common denominator: \(\frac{7\times 3yx}{6x\times 3yx}-\frac{2\times 2x\times x}{9y\times 2x\times x}+\frac{2\times 3y}{3x^{2}\times 2\times 3y}\) - step2: Multiply the numbers: \(\frac{7\times 3yx}{18xyx}-\frac{2\times 2x\times x}{9y\times 2x\times x}+\frac{2\times 3y}{3x^{2}\times 2\times 3y}\) - step3: Multiply the numbers: \(\frac{7\times 3yx}{18xyx}-\frac{2\times 2x\times x}{18yx\times x}+\frac{2\times 3y}{3x^{2}\times 2\times 3y}\) - step4: Multiply the terms: \(\frac{7\times 3yx}{18xyx}-\frac{2\times 2x\times x}{18yx\times x}+\frac{2\times 3y}{18x^{2}y}\) - step5: Multiply the terms: \(\frac{7\times 3yx}{18x^{2}y}-\frac{2\times 2x\times x}{18yx\times x}+\frac{2\times 3y}{18x^{2}y}\) - step6: Multiply the terms: \(\frac{7\times 3yx}{18x^{2}y}-\frac{2\times 2x\times x}{18yx^{2}}+\frac{2\times 3y}{18x^{2}y}\) - step7: Rewrite the expression: \(\frac{7\times 3yx}{18x^{2}y}-\frac{2\times 2x\times x}{18x^{2}y}+\frac{2\times 3y}{18x^{2}y}\) - step8: Transform the expression: \(\frac{7\times 3yx-2\times 2x\times x+2\times 3y}{18x^{2}y}\) - step9: Multiply the terms: \(\frac{21yx-2\times 2x\times x+2\times 3y}{18x^{2}y}\) - step10: Multiply the terms: \(\frac{21yx-4x^{2}+2\times 3y}{18x^{2}y}\) - step11: Multiply the terms: \(\frac{21yx-4x^{2}+6y}{18x^{2}y}\) - step12: Simplify: \(\frac{21yx-4x^{2}+6y}{18yx^{2}}\) Calculate or simplify the expression \( 1 + 2 - (2*a - 1)/2 \). Simplify the expression by following steps: - step0: Solution: \(1+2-\frac{\left(2a-1\right)}{2}\) - step1: Remove the parentheses: \(1+2-\frac{2a-1}{2}\) - step2: Add the numbers: \(3-\frac{2a-1}{2}\) - step3: Reduce fractions to a common denominator: \(\frac{3\times 2}{2}-\frac{2a-1}{2}\) - step4: Transform the expression: \(\frac{3\times 2-\left(2a-1\right)}{2}\) - step5: Multiply the numbers: \(\frac{6-\left(2a-1\right)}{2}\) - step6: Subtract the terms: \(\frac{7-2a}{2}\) Calculate or simplify the expression \( 5/(6*x*y) - (x + 2)/(2*x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{5}{6xy}-\frac{\left(x+2\right)}{2x}\) - step1: Remove the parentheses: \(\frac{5}{6xy}-\frac{x+2}{2x}\) - step2: Reduce fractions to a common denominator: \(\frac{5}{6xy}-\frac{\left(x+2\right)\times 3y}{2x\times 3y}\) - step3: Multiply the numbers: \(\frac{5}{6xy}-\frac{\left(x+2\right)\times 3y}{6xy}\) - step4: Transform the expression: \(\frac{5-\left(x+2\right)\times 3y}{6xy}\) - step5: Multiply the terms: \(\frac{5-\left(3xy+6y\right)}{6xy}\) - step6: Remove the parentheses: \(\frac{5-3xy-6y}{6xy}\) Calculate or simplify the expression \( (x - 3)/3 - (x - 2)/2 + (x + 1)/6 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-3\right)}{3}-\frac{\left(x-2\right)}{2}+\frac{\left(x+1\right)}{6}\) - step1: Remove the parentheses: \(\frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(x-3\right)\times 2}{3\times 2}-\frac{\left(x-2\right)\times 3}{2\times 3}+\frac{x+1}{6}\) - step3: Multiply the numbers: \(\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{2\times 3}+\frac{x+1}{6}\) - step4: Multiply the numbers: \(\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{6}+\frac{x+1}{6}\) - step5: Transform the expression: \(\frac{\left(x-3\right)\times 2-\left(x-2\right)\times 3+x+1}{6}\) - step6: Multiply the terms: \(\frac{2x-6-\left(x-2\right)\times 3+x+1}{6}\) - step7: Multiply the terms: \(\frac{2x-6-\left(3x-6\right)+x+1}{6}\) - step8: Calculate: \(\frac{1}{6}\) Here are the simplified results for each expression: 1. **Expression:** \( \frac{2x}{3} + \frac{1}{6} \) **Simplified Result:** \( \frac{4x + 1}{6} \) 2. **Expression:** \( \frac{7}{6x} - \frac{2}{9y} + \frac{1}{3x^2} \) **Simplified Result:** \( \frac{21yx - 4x^2 + 6y}{18yx^2} \) 3. **Expression:** \( \frac{3}{x} + \frac{2}{x^2} \) **Simplified Result:** \( \frac{3x + 2}{x^2} \) 4. **Expression:** \( 1 + 2 - \frac{2a - 1}{2} \) **Simplified Result:** \( \frac{7 - 2a}{2} \) 5. **Expression:** \( \frac{5}{6xy} - \frac{x + 2}{2x} \) **Simplified Result:** \( \frac{5 - 3xy - 6y}{6xy} \) 6. **Expression:** \( \frac{5}{2x} - \frac{x + 1}{x - 2} \) **Simplified Result:** \( \frac{3x - 10 - 2x^2}{2x^2 - 4x} \) 7. **Expression:** \( \frac{2}{x} + \frac{5}{x + 1} \) **Simplified Result:** \( \frac{7x + 2}{x^2 + x} \) 8. **Expression:** \( \frac{x + 2}{x^2 + 2} - \frac{6}{x + 2} \) **Simplified Result:** \( \frac{-5x^2 + 4x - 8}{x^3 + 2x^2 + 2x + 4} \) 9. **Expression:** \( \frac{x - 3}{x + 3} - \frac{x + 3}{x - 3} \) **Simplified Result:** \( -\frac{12x}{x^2 - 9} \) 10. **Expression:** \( \frac{4}{(2x + 1)^2} - \frac{x + 1}{2x + 1} \) **Simplified Result:** \( \frac{3 - 2x^2 - 3x}{4x^2 + 4x + 1} \) 11. **Expression:** \( \frac{2}{2x} + \frac{7x + 1}{4x^2y} - \frac{5xy}{4} + 1 \) **Simplified Result:** \( \frac{4xy + 7x + 1 - 5x^3y^2 + 4x^2y}{4yx^2} \) 12. **Expression:** \( \frac{x - 3}{3} - \frac{x - 2}{2} + \frac{x + 1}{6} \) **Simplified Result:** \( \frac{1}{6} \) (or \( 0.1\dot{6} \)) 13. **Expression:** \( x - \frac{2x}{3x - 2} \) **Simplified Result:** \( \frac{3x^2 - 4x}{3x - 2} \) If you need further assistance or explanations for any specific step, feel free to ask!