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\( A=a-\frac{1}{2-\frac{2}{2-\frac{2}{a}}} \quad M= \) Express \( A \) as a single fraction in its simplest form

Ask by Rogers Carrillo. in the United Kingdom
Mar 13,2025

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Tutor-Verified Answer

Answer

\( A = \frac{a^2 - 3a + 1}{a - 2} \)

Solution

Calculate or simplify the expression \( a-\frac{1}{2-\frac{2}{2-\frac{2}{a}}} \). Simplify the expression by following steps: - step0: Solution: \(a-\frac{1}{2-\frac{2}{2-\frac{2}{a}}}\) - step1: Subtract the terms: \(a-\frac{1}{2-\frac{2}{\frac{2a-2}{a}}}\) - step2: Divide the terms: \(a-\frac{1}{2-\frac{a}{a-1}}\) - step3: Subtract the terms: \(a-\frac{1}{\frac{a-2}{a-1}}\) - step4: Divide the terms: \(a-\frac{a-1}{a-2}\) - step5: Reduce fractions to a common denominator: \(\frac{a\left(a-2\right)}{a-2}-\frac{a-1}{a-2}\) - step6: Transform the expression: \(\frac{a\left(a-2\right)-\left(a-1\right)}{a-2}\) - step7: Multiply the terms: \(\frac{a^{2}-2a-\left(a-1\right)}{a-2}\) - step8: Subtract the terms: \(\frac{a^{2}-3a+1}{a-2}\) The expression \( A \) can be simplified to \( \frac{a^2 - 3a + 1}{a - 2} \).

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To express \( A \) as a single fraction, let's start by simplifying the expression inside \( A \). Given: \[ A = a - \frac{1}{2 - \frac{2}{2 - \frac{2}{a}}} \] First, simplify the innermost fraction \( 2 - \frac{2}{a} \): \[ 2 - \frac{2}{a} = \frac{2a - 2}{a} = \frac{2(a - 1)}{a} \] Now substitute this back into the middle fraction: \[ 2 - \frac{2}{2 - \frac{2}{a}} = 2 - \frac{2}{\frac{2(a - 1)}{a}} = 2 - \frac{2a}{2(a - 1)} = 2 - \frac{a}{a - 1} \] To combine this, find a common denominator: \[ 2 - \frac{a}{a - 1} = \frac{2(a - 1) - a}{a - 1} = \frac{2a - 2 - a}{a - 1} = \frac{a - 2}{a - 1} \] Now substitute this back into the original equation for \( A \): \[ A = a - \frac{1}{\frac{a - 2}{a - 1}} = a - \frac{a - 1}{a - 2} \] Now, to combine these terms, we need a common denominator: \[ A = \frac{A(a - 2) - (a - 1)}{a - 2} = \frac{Aa - 2A - a + 1}{a - 2} \] Notice the mistake. Let's go back to: \[ A = a - \frac{a - 1}{a - 2} = \frac{a(a - 2) - (a - 1)}{a - 2} = \frac{a^2 - 2a - a + 1}{a - 2} = \frac{a^2 - 3a + 1}{a - 2} \] So, the final expression for \( A \) simplifies to: \[ A = \frac{a^2 - 3a + 1}{a - 2} \]

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