To rewrite the expression \(\frac{18r}{r^2 - r - 12}\) with the indicated denominator \((r - 4)(r + 3)(r + 2)\), we first need to factor \(r^2 - r - 12\).
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Factor the denominator: We are looking for two numbers that multiply to \(-12\) (the constant term) and add to \(-1\) (the coefficient of \(r\)). Those numbers are \(-4\) and \(3\).
So, we can factor: \[ r^2 - r - 12 = (r - 4)(r + 3) \]
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Rewrite the original rational expression: Now we can express the original rational expression: \[ \frac{18r}{r^2 - r - 12} = \frac{18r}{(r - 4)(r + 3)} \]
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Set the equal sign with the new denominator: To adjust it to have the new denominator of \((r - 4)(r + 3)(r + 2)\), we can multiply the numerator and denominator of \(\frac{18r}{(r - 4)(r + 3)}\) by \((r + 2)\): \[ \frac{18r}{(r - 4)(r + 3)} \cdot \frac{(r + 2)}{(r + 2)} = \frac{18r(r + 2)}{(r - 4)(r + 3)(r + 2)} \]
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Write the final expression: This gives us the expression with the specified denominator: \[ \frac{18r(r + 2)}{(r - 4)(r + 3)(r + 2)} \]
Thus, the rational expression as an equivalent expression with the indicated denominator is: \[ \frac{18r(r + 2)}{(r - 4)(r + 3)(r + 2)} \] So, the answer is: \[ 18r(r + 2) \]