Exercise 5: Simplify the following (you may assume that denominators are non-zero): \( \begin{array}{lll}\text { a) } 2-\frac{1}{2 x^{2}}+\frac{3}{x} & \text { c) } \frac{8 a^{2}+4 a}{4 a} & \text { e) } \frac{2}{3}+\frac{3 x+2}{x}-\frac{x+4}{2 x} \\ \text { b) } \frac{5}{6}-\frac{4 q}{p}+\frac{4 p}{q} & \text { d) } \frac{5 x+2}{5}-\frac{3 x+10}{3} & \text { f) } \frac{2 x+3}{x}+\frac{7}{4 x^{2}}-\frac{4 x+1}{3}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(2-\frac{1}{2x^{2}}+\frac{3}{x}\) - step1: Reduce fractions to a common denominator: \(\frac{2\times 2x^{2}}{2x^{2}}-\frac{1}{2x^{2}}+\frac{3\times 2x}{x\times 2x}\) - step2: Reorder the terms: \(\frac{2\times 2x^{2}}{2x^{2}}-\frac{1}{2x^{2}}+\frac{3\times 2x}{2x\times x}\) - step3: Multiply the terms: \(\frac{2\times 2x^{2}}{2x^{2}}-\frac{1}{2x^{2}}+\frac{3\times 2x}{2x^{2}}\) - step4: Transform the expression: \(\frac{2\times 2x^{2}-1+3\times 2x}{2x^{2}}\) - step5: Multiply the terms: \(\frac{4x^{2}-1+3\times 2x}{2x^{2}}\) - step6: Multiply the terms: \(\frac{4x^{2}-1+6x}{2x^{2}}\) Calculate or simplify the expression \( (2*x + 3)/x + (7/(4*x^2)) - ((4*x + 1)/3) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2x+3\right)}{x}+\frac{7}{4x^{2}}-\left(\frac{\left(4x+1\right)}{3}\right)\) - step1: Remove the parentheses: \(\frac{2x+3}{x}+\frac{7}{4x^{2}}-\left(\frac{4x+1}{3}\right)\) - step2: Remove the parentheses: \(\frac{2x+3}{x}+\frac{7}{4x^{2}}-\frac{4x+1}{3}\) - step3: Reduce fractions to a common denominator: \(\frac{\left(2x+3\right)\times 4\times 3x}{x\times 4\times 3x}+\frac{7\times 3}{4x^{2}\times 3}-\frac{\left(4x+1\right)\times 4x\times x}{3\times 4x\times x}\) - step4: Multiply the numbers: \(\frac{\left(2x+3\right)\times 4\times 3x}{12x\times x}+\frac{7\times 3}{4x^{2}\times 3}-\frac{\left(4x+1\right)\times 4x\times x}{3\times 4x\times x}\) - step5: Multiply the numbers: \(\frac{\left(2x+3\right)\times 4\times 3x}{12x\times x}+\frac{7\times 3}{12x^{2}}-\frac{\left(4x+1\right)\times 4x\times x}{3\times 4x\times x}\) - step6: Multiply the numbers: \(\frac{\left(2x+3\right)\times 4\times 3x}{12x\times x}+\frac{7\times 3}{12x^{2}}-\frac{\left(4x+1\right)\times 4x\times x}{12x\times x}\) - step7: Multiply the terms: \(\frac{\left(2x+3\right)\times 4\times 3x}{12x^{2}}+\frac{7\times 3}{12x^{2}}-\frac{\left(4x+1\right)\times 4x\times x}{12x\times x}\) - step8: Multiply the terms: \(\frac{\left(2x+3\right)\times 4\times 3x}{12x^{2}}+\frac{7\times 3}{12x^{2}}-\frac{\left(4x+1\right)\times 4x\times x}{12x^{2}}\) - step9: Transform the expression: \(\frac{\left(2x+3\right)\times 4\times 3x+7\times 3-\left(4x+1\right)\times 4x\times x}{12x^{2}}\) - step10: Multiply the terms: \(\frac{24x^{2}+36x+7\times 3-\left(4x+1\right)\times 4x\times x}{12x^{2}}\) - step11: Multiply the numbers: \(\frac{24x^{2}+36x+21-\left(4x+1\right)\times 4x\times x}{12x^{2}}\) - step12: Multiply the terms: \(\frac{24x^{2}+36x+21-\left(16x^{3}+4x^{2}\right)}{12x^{2}}\) - step13: Calculate: \(\frac{20x^{2}+36x+21-16x^{3}}{12x^{2}}\) Calculate or simplify the expression \( (2/3) + ((3*x + 2)/x) - ((x + 4)/(2*x)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{2}{3}+\left(\frac{\left(3x+2\right)}{x}\right)-\left(\frac{\left(x+4\right)}{2x}\right)\) - step1: Remove the parentheses: \(\frac{2}{3}+\left(\frac{3x+2}{x}\right)-\left(\frac{x+4}{2x}\right)\) - step2: Remove the parentheses: \(\frac{2}{3}+\frac{3x+2}{x}-\left(\frac{x+4}{2x}\right)\) - step3: Remove the parentheses: \(\frac{2}{3}+\frac{3x+2}{x}-\frac{x+4}{2x}\) - step4: Reduce fractions to a common denominator: \(\frac{2\times 2x}{3\times 2x}+\frac{\left(3x+2\right)\times 3\times 2}{x\times 3\times 2}-\frac{\left(x+4\right)\times 3}{2x\times 3}\) - step5: Multiply the numbers: \(\frac{2\times 2x}{6x}+\frac{\left(3x+2\right)\times 3\times 2}{x\times 3\times 2}-\frac{\left(x+4\right)\times 3}{2x\times 3}\) - step6: Multiply the numbers: \(\frac{2\times 2x}{6x}+\frac{\left(3x+2\right)\times 3\times 2}{6x}-\frac{\left(x+4\right)\times 3}{2x\times 3}\) - step7: Multiply the numbers: \(\frac{2\times 2x}{6x}+\frac{\left(3x+2\right)\times 3\times 2}{6x}-\frac{\left(x+4\right)\times 3}{6x}\) - step8: Transform the expression: \(\frac{2\times 2x+\left(3x+2\right)\times 3\times 2-\left(x+4\right)\times 3}{6x}\) - step9: Multiply the terms: \(\frac{4x+\left(3x+2\right)\times 3\times 2-\left(x+4\right)\times 3}{6x}\) - step10: Multiply the terms: \(\frac{4x+18x+12-\left(x+4\right)\times 3}{6x}\) - step11: Multiply the terms: \(\frac{4x+18x+12-\left(3x+12\right)}{6x}\) - step12: Calculate: \(\frac{19x}{6x}\) - step13: Reduce the fraction: \(\frac{19}{6}\) Calculate or simplify the expression \( (5/6) - (4*q/p) + (4*p/q) \). Simplify the expression by following steps: - step0: Solution: \(\frac{5}{6}-\frac{4q}{p}+\frac{4p}{q}\) - step1: Reduce fractions to a common denominator: \(\frac{5pq}{6pq}-\frac{4q\times 6q}{p\times 6q}+\frac{4p\times 6p}{q\times 6p}\) - step2: Reorder the terms: \(\frac{5pq}{6pq}-\frac{4q\times 6q}{6pq}+\frac{4p\times 6p}{q\times 6p}\) - step3: Reorder the terms: \(\frac{5pq}{6pq}-\frac{4q\times 6q}{6pq}+\frac{4p\times 6p}{6qp}\) - step4: Rewrite the expression: \(\frac{5pq}{6pq}-\frac{4q\times 6q}{6pq}+\frac{4p\times 6p}{6pq}\) - step5: Transform the expression: \(\frac{5pq-4q\times 6q+4p\times 6p}{6pq}\) - step6: Multiply the terms: \(\frac{5pq-24q^{2}+4p\times 6p}{6pq}\) - step7: Multiply the terms: \(\frac{5pq-24q^{2}+24p^{2}}{6pq}\) Calculate or simplify the expression \( (5*x + 2)/5 - (3*x + 10)/3 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(5x+2\right)}{5}-\frac{\left(3x+10\right)}{3}\) - step1: Remove the parentheses: \(\frac{5x+2}{5}-\frac{3x+10}{3}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(5x+2\right)\times 3}{5\times 3}-\frac{\left(3x+10\right)\times 5}{3\times 5}\) - step3: Multiply the numbers: \(\frac{\left(5x+2\right)\times 3}{15}-\frac{\left(3x+10\right)\times 5}{3\times 5}\) - step4: Multiply the numbers: \(\frac{\left(5x+2\right)\times 3}{15}-\frac{\left(3x+10\right)\times 5}{15}\) - step5: Transform the expression: \(\frac{\left(5x+2\right)\times 3-\left(3x+10\right)\times 5}{15}\) - step6: Multiply the terms: \(\frac{15x+6-\left(3x+10\right)\times 5}{15}\) - step7: Multiply the terms: \(\frac{15x+6-\left(15x+50\right)}{15}\) - step8: Calculate: \(\frac{-44}{15}\) - step9: Rewrite the fraction: \(-\frac{44}{15}\) Calculate or simplify the expression \( (8*a^2 + 4*a)/(4*a) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(8a^{2}+4a\right)}{4a}\) - step1: Remove the parentheses: \(\frac{8a^{2}+4a}{4a}\) - step2: Factor: \(\frac{4a\left(2a+1\right)}{4a}\) - step3: Reduce the fraction: \(2a+1\) Let's simplify each expression step by step: ### a) \( 2 - \frac{1}{2x^2} + \frac{3}{x} \) 1. **Common Denominator**: The common denominator for the terms is \( 2x^2 \). 2. **Rewrite Each Term**: - \( 2 = \frac{4x^2}{2x^2} \) - \( -\frac{1}{2x^2} \) remains the same. - \( \frac{3}{x} = \frac{6x}{2x^2} \) 3. **Combine**: \[ \frac{4x^2 - 1 + 6x}{2x^2} \] **Final Result**: \( \frac{4x^2 + 6x - 1}{2x^2} \) --- ### b) \( \frac{5}{6} - \frac{4q}{p} + \frac{4p}{q} \) 1. **Common Denominator**: The common denominator is \( 6pq \). 2. **Rewrite Each Term**: - \( \frac{5}{6} = \frac{5pq}{6pq} \) - \( -\frac{4q}{p} = -\frac{24q^2}{6pq} \) - \( \frac{4p}{q} = \frac{24p^2}{6pq} \) 3. **Combine**: \[ \frac{5pq - 24q^2 + 24p^2}{6pq} \] **Final Result**: \( \frac{5pq - 24q^2 + 24p^2}{6pq} \) --- ### c) \( \frac{8a^2 + 4a}{4a} \) 1. **Simplify**: \[ \frac{8a^2}{4a} + \frac{4a}{4a} = 2a + 1 \] **Final Result**: \( 2a + 1 \) --- ### d) \( \frac{5x + 2}{5} - \frac{3x + 10}{3} \) 1. **Common Denominator**: The common denominator is \( 15 \). 2. **Rewrite Each Term**: - \( \frac{5x + 2}{5} = \frac{3(5x + 2)}{15} = \frac{15x + 6}{15} \) - \( -\frac{3x + 10}{3} = -\frac{5(3x + 10)}{15} = -\frac{15x + 50}{15} \) 3. **Combine**: \[ \frac{15x + 6 - 15x - 50}{15} = \frac{-44}{15} \] **Final Result**: \( -\frac{44}{15} \) --- ### e) \( \frac{2}{3} + \frac{3x + 2}{x} - \frac{x + 4}{2x} \) 1. **Common Denominator**: The common denominator is \( 6x \). 2. **Rewrite Each Term**: - \( \frac{2}{3} = \frac{4x}{6x} \) - \( \frac{3x + 2}{x} = \frac{6(3x + 2)}{6x} = \frac{18x + 12}{6x} \) - \( -\frac{x + 4}{2x} = -\frac{3(x + 4)}{6x} = -\frac{3x + 12}{6x} \) 3. **Combine**: \[ \frac{4x + 18x + 12 - 3x - 12}{6x} = \frac{19x}{6x} = \frac{19}{6} \] **Final Result**: \( \frac{19}{6} \) --- ### f) \( \frac{2x + 3}{x} + \frac{7}{4x^2} - \frac{4x + 1}{3} \) 1. **Common Denominator**: The common denominator is \( 12x^2 \). 2. **Rewrite Each Term**: - \( \frac{2x + 3}{x} = \frac{12(2x + 3)}{12x^2} = \frac{24x + 36}{12x^2} \) - \( \frac{7}{4x^2} = \frac{21}{12x^2} \) - \( -\frac{4x + 1}{3} = -\frac{4(4x + 1)}{12x^2} = -\frac{16x + 4}{12x^2} \) 3. **Combine**: \[ \frac{24x + 36 + 21 - 16x - 4}{12x^2} = \frac{8x + 53}{12x^2} \] **Final Result**: \( \frac{8x + 53}{12x^2} \) --- ### Summary of Results: - a) \( \frac{4x^2 + 6x - 1}{2x^2} \) - b) \( \frac{5pq - 24q^2 + 24p^2}{6pq} \) - c) \( 2a + 1 \) - d) \( -\frac{44}{15} \) - e) \( \frac{19}{6} \) - f) \( \frac{8x + 53}{12x^2} \)