To simplify the expression \(\frac{\frac{1}{9} - \frac{1}{m^2}}{\frac{1}{3} + \frac{1}{m}}\), we will first find a common denominator for both the numerator and the denominator.
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Numerator: \(\frac{1}{9} - \frac{1}{m^2}\)
The common denominator for \(9\) and \(m^2\) is \(9m^2\). Thus, we rewrite the terms as: \[ \frac{1}{9} = \frac{m^2}{9m^2} \] \[ \frac{1}{m^2} = \frac{9}{9m^2} \] Hence, the numerator becomes: \[ \frac{m^2}{9m^2} - \frac{9}{9m^2} = \frac{m^2 - 9}{9m^2} \]
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Denominator: \(\frac{1}{3} + \frac{1}{m}\)
The common denominator for \(3\) and \(m\) is \(3m\). Rewriting the terms, we have: \[ \frac{1}{3} = \frac{m}{3m} \] \[ \frac{1}{m} = \frac{3}{3m} \] So, the denominator becomes: \[ \frac{m}{3m} + \frac{3}{3m} = \frac{m + 3}{3m} \]
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Putting it all together:
Now we have: \[ \frac{\frac{m^2 - 9}{9m^2}}{\frac{m + 3}{3m}} \] This simplifies to: \[ \frac{m^2 - 9}{9m^2} \cdot \frac{3m}{m + 3} = \frac{3m(m^2 - 9)}{9m^2(m + 3)} \]
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Further simplification:
Notice that \(m^2 - 9\) can be factored as \((m - 3)(m + 3)\): \[ \frac{3m(m - 3)(m + 3)}{9m^2(m + 3)} \] Now, we can cancel \((m + 3)\) from the numerator and denominator (assuming \(m \neq -3\)): \[ = \frac{3m(m - 3)}{9m^2} \] This can be simplified further: \[ = \frac{3(m - 3)}{9m} = \frac{m - 3}{3m} \]
Thus, the simplified expression is: \[ \frac{m - 3}{3m} \] (where \(m \neq 0\) and \(m \neq -3\)).