Which expression is equivalent to the following complex fraction? \( \frac{\frac{x}{x-3}}{\frac{x^{2}}{x^{2}-9}} \)
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To simplify the complex fraction \( \frac{\frac{x}{x-3}}{\frac{x^{2}}{x^{2}-9}} \), you can multiply by the reciprocal of the denominator. This gives us: \[ \frac{x}{x-3} \times \frac{x^{2}-9}{x^{2}}. \] Now, recognize that \( x^{2}-9 \) can be factored using the difference of squares: \[ x^{2}-9 = (x-3)(x+3). \] So, the expression becomes: \[ \frac{x \cdot (x-3)(x+3)}{(x-3)(x^{2})}. \] You can cancel \( (x-3) \) from the numerator and denominator (as long as \( x \neq 3 \)), leading to: \[ \frac{x(x+3)}{x^{2}}. \] This simplifies to: \[ \frac{x + 3}{x} \quad \text{for } x \neq 3. \] Thus, the expression \( \frac{x + 3}{x} \) is the simplified version of the given complex fraction.