Pregunta
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Simplify. a) \( \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} \) b) \( \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} \) c) \( \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} \) d) \( \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{x^{3}-2 x^{2}}{x^{2}-16} \times \frac{1}{x+4} \) e) \( \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \) f) \( \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \)

Ask by Mccarthy Erickson. in South Africa
Feb 21,2025

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a) \( \frac{b}{a} \) b) \( \frac{9}{x^{2}+6x+9} \) c) \( 6x^{2} \) d) \( \frac{x+3}{3x^{3}} \) e) \( 1 \) f) \( \frac{a^{2}-2a+1}{a^{2}} \)

Solución

Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(ab-a^{2}\right)}{\left(b^{2}-a^{2}\right)}}{\left(\frac{a^{2}}{\left(b^{2}+ab\right)}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{ab-a^{2}}{b^{2}-a^{2}}}{\frac{a^{2}}{b^{2}+ab}}\) - step2: Divide the terms: \(\frac{\frac{a}{b+a}}{\frac{a^{2}}{b^{2}+ab}}\) - step3: Multiply by the reciprocal: \(\frac{a}{b+a}\times \frac{b^{2}+ab}{a^{2}}\) - step4: Reduce the fraction: \(\frac{1}{b+a}\times \frac{b^{2}+ab}{a}\) - step5: Rewrite the expression: \(\frac{1}{b+a}\times \frac{b\left(b+a\right)}{a}\) - step6: Reduce the fraction: \(1\times \frac{b}{a}\) - step7: Multiply the terms: \(\frac{b}{a}\) Calculate or simplify the expression \( (2*x^2-6*x)/(x^2-1) / ((x-3)/(x^2-x)) / (1/(3*x+3)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\frac{\left(2x^{2}-6x\right)}{\left(x^{2}-1\right)}}{\left(\frac{\left(x-3\right)}{\left(x^{2}-x\right)}\right)}}{\left(\frac{1}{\left(3x+3\right)}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{\frac{2x^{2}-6x}{x^{2}-1}}{\frac{x-3}{x^{2}-x}}}{\frac{1}{3x+3}}\) - step2: Divide the terms: \(\frac{\frac{2x^{2}}{x+1}}{\frac{1}{3x+3}}\) - step3: Multiply by the reciprocal: \(\frac{2x^{2}}{x+1}\times \left(3x+3\right)\) - step4: Rewrite the expression: \(\frac{2x^{2}}{x+1}\times 3\left(x+1\right)\) - step5: Reduce the fraction: \(2x^{2}\times 3\) - step6: Multiply the terms: \(6x^{2}\) Calculate or simplify the expression \( (3*x^2+27)/(2*x+6) * (6*x-18)/(x^4-81) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(3x^{2}+27\right)}{\left(2x+6\right)}\times \left(6x-18\right)}{\left(x^{4}-81\right)}\) - step1: Remove the parentheses: \(\frac{\frac{3x^{2}+27}{2x+6}\times \left(6x-18\right)}{x^{4}-81}\) - step2: Multiply the terms: \(\frac{\frac{\left(3x^{2}+27\right)\left(3x-9\right)}{x+3}}{x^{4}-81}\) - step3: Multiply by the reciprocal: \(\frac{\left(3x^{2}+27\right)\left(3x-9\right)}{x+3}\times \frac{1}{x^{4}-81}\) - step4: Rewrite the expression: \(\frac{3\left(x^{2}+9\right)\left(3x-9\right)}{x+3}\times \frac{1}{\left(x^{2}+9\right)\left(x^{2}-9\right)}\) - step5: Reduce the fraction: \(\frac{3\left(3x-9\right)}{x+3}\times \frac{1}{x^{2}-9}\) - step6: Rewrite the expression: \(\frac{3\times 3\left(x-3\right)}{x+3}\times \frac{1}{\left(x-3\right)\left(x+3\right)}\) - step7: Reduce the fraction: \(\frac{3\times 3}{x+3}\times \frac{1}{x+3}\) - step8: Multiply the terms: \(\frac{9}{\left(x+3\right)\left(x+3\right)}\) - step9: Multiply the terms: \(\frac{9}{\left(x+3\right)^{2}}\) - step10: Calculate: \(\frac{9}{x^{2}+6x+9}\) Calculate or simplify the expression \( (a^2-2*a+1)/(a^2+2*a+1) * (a^2-1)/(a^2) * (a^2+a)/(a^2-a) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\frac{\left(a^{2}-2a+1\right)}{\left(a^{2}+2a+1\right)}\times \left(a^{2}-1\right)}{a^{2}}\times \left(a^{2}+a\right)}{\left(a^{2}-a\right)}\) - step1: Remove the parentheses: \(\frac{\frac{\frac{a^{2}-2a+1}{a^{2}+2a+1}\times \left(a^{2}-1\right)}{a^{2}}\times \left(a^{2}+a\right)}{a^{2}-a}\) - step2: Multiply the terms: \(\frac{\frac{\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a+1}}{a^{2}}\times \left(a^{2}+a\right)}{a^{2}-a}\) - step3: Divide the terms: \(\frac{\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a^{2}\left(a+1\right)}\times \left(a^{2}+a\right)}{a^{2}-a}\) - step4: Multiply the terms: \(\frac{\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a}}{a^{2}-a}\) - step5: Multiply by the reciprocal: \(\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a}\times \frac{1}{a^{2}-a}\) - step6: Rewrite the expression: \(\frac{\left(a-1\right)^{2}\left(a-1\right)}{a}\times \frac{1}{a\left(a-1\right)}\) - step7: Reduce the fraction: \(\frac{\left(a-1\right)\left(a-1\right)}{a}\times \frac{1}{a}\) - step8: Multiply the terms: \(\frac{\left(a-1\right)^{2}}{a\times a}\) - step9: Multiply the terms: \(\frac{\left(a-1\right)^{2}}{a^{2}}\) - step10: Calculate: \(\frac{a^{2}-2a+1}{a^{2}}\) Calculate or simplify the expression \( (x^2+x-6)/(3*x^2-12*x) / ((x^3-2*x^2)/(x^2-16)) * (1/(x+4)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{2}+x-6\right)}{\left(3x^{2}-12x\right)}}{\left(\frac{\left(x^{3}-2x^{2}\right)}{\left(x^{2}-16\right)}\right)}\times \left(\frac{1}{\left(x+4\right)}\right)\) - step1: Remove the parentheses: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{x^{3}-2x^{2}}{x^{2}-16}}\times \left(\frac{1}{x+4}\right)\) - step2: Remove the parentheses: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{x^{3}-2x^{2}}{x^{2}-16}}\times \frac{1}{x+4}\) - step3: Divide the terms: \(\frac{\left(x+3\right)\left(x+4\right)}{3x^{3}}\times \frac{1}{x+4}\) - step4: Reduce the fraction: \(\frac{x+3}{3x^{3}}\times 1\) - step5: Multiply the terms: \(\frac{x+3}{3x^{3}}\) Calculate or simplify the expression \( (a^2*b-a*b)/(a^3+a^2) / ((a^2*b-2*a*b+b)/(a^3-a)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(a^{2}b-ab\right)}{\left(a^{3}+a^{2}\right)}}{\left(\frac{\left(a^{2}b-2ab+b\right)}{\left(a^{3}-a\right)}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{a^{2}b-ab}{a^{3}+a^{2}}}{\frac{a^{2}b-2ab+b}{a^{3}-a}}\) - step2: Divide the terms: \(\frac{\frac{ab-b}{a^{2}+a}}{\frac{a^{2}b-2ab+b}{a^{3}-a}}\) - step3: Divide the terms: \(\frac{\frac{ab-b}{a^{2}+a}}{\frac{ab-b}{a^{2}+a}}\) - step4: Divide the terms: \(1\) Here are the simplified results for each part of the problem: a) \[ \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} = \frac{b}{a} \] b) \[ \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} = \frac{9}{x^{2}+6x+9} \] c) \[ \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} = 6x^{2} \] d) \[ \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{x^{3}-2 x^{2}}{x^{2}-16} \times \frac{1}{x+4} = \frac{x+3}{3x^{3}} \] e) \[ \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} = 1 \] f) \[ \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} = \frac{a^{2}-2a+1}{a^{2}} \] If you need further explanations or steps for any of these simplifications, feel free to ask!

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Extra Insights

Let’s dive into these simplifications and have some fun while unraveling these algebraic expressions! a) To simplify \( \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} \), we start with the first expression. We can factor \( a(b-a) \) in the numerator and \( (b+a)(b-a) \) in the denominator, leading to \( \frac{a(b-a)}{(b+a)(b-a)} \). The terms \( (b-a) \) cancel, giving us \( \frac{a}{b+a} \). Now, we flip the second fraction and multiply: \( \frac{a}{b+a} \cdot \frac{b^{2}+ab}{a^{2}} \). So, simplify to \( \frac{a(b^{2}+ab)}{a^{2}(b+a)} \), further simplifying gives \( \frac{b^{2}+ab}{a(b+a)} \). b) For \( \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} \), we can factor \( 3(x^{2}+9) \) and \( 2(x+3) \) in the first expression and \( 6(x-3) \) in the second. The denominator \( x^{4}-81 \) factors to \( (x^{2}-9)(x^{2}+9) \) and further to \( (x-3)(x+3)(x^{2}+9) \). Cancel \( (x-3) \) and \( (x+3) \), leading to \( \frac{3 \cdot 6}{2} \), which simplifies to \( 9 \). c) For \( \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} \), we factor \( 2x(x-3) \) and \( (x-1)(x+1) \) in the first expression, while the second is \( (x-3)/(x(x-1)) \). The complex division will involve reciprocating the last expression, resulting in some cancellations. After all reductions, you're left with \( \frac{6(x-3)}{(x+1)(3)} \) leading you to the simplified form \( \frac{2(x-3)}{(x+1)} \). d) When simplifying \( \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{x^{3}-2 x^{2}}{x^{2}-16} \times \frac{1}{x+4} \), factor to \( \frac{(x-2)(x+3)}{3x(x-4)} \) for the first and \( \frac{x^{2}(x-2)}{(x-4)(x+4)} \) for the second. Flipping and multiplying gives you a chance to cancel \( (x-2) \) and leads to simplifications that ultimately yield \( \frac{x+3}{3x} \). e) For \( \frac{a^{2}b-ab}{a^{3}+a^{2}} \div \frac{a^{2}b-2ab+b}{a^{3}-a} \), you can factor \( ab(a-b) \) and \( a^{2}(a+1) \) in the first and adjust the second expression to something similar. After simplifications, flipping the second fraction, you're left with a nice product giving you clear insights on how the 'a's and 'b's play along. f) Finally, in \( \frac{a^{2}-2a+1}{a^{2}+2a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \), factor the first to \( \frac{(a-1)^{2}}{(a+1)^{2}} \), the second to \( \frac{(a-1)(a+1)}{a^2} \) and the third simplifies to \( \frac{a(a+1)}{a(a-1)} \). These lead to valuable cancellations that empower students to grasp fractions better. Enjoy the math journey and keep practicing!

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