Simplify the complex fraction. (frac (x)/(3)-(3)/(x))((1)/(3)+(1)/(x))

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$Simplify the complex fraction. (frac (x)/(3)-(3)/(x))((1)/(3)+(1)/(x))$

Simplify the complex fraction. (frac (x)/(3)-(3)/(x))((1)/(3)+(1)/(x))

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## Step 1: Simplify the numerator The numerator of the complex fraction is $\frac {x}{3}-\frac {3}{x}$. To simplify this, find a common denominator, which is 3x in this case. ### $\frac {x}{3}-\frac {3}{x} = \frac {x^2}{3x} - \frac {9}{3x} = \frac {x^2 - 9}{3x}$ ## Step 2: Simplify the denominator The denominator of the complex fraction is $\frac {1}{3}+\frac {1}{x}$. To simplify this, find a common denominator, which is 3x in this case. ### $\frac {1}{3}+\frac {1}{x} = \frac {x}{3x} + \frac {3}{3x} = \frac {x + 3}{3x}$ ## Step 3: Simplify the complex fraction The complex fraction is now $\frac {\frac {x^2 - 9}{3x}}{\frac {x + 3}{3x}}$. To simplify this, multiply the numerator by the reciprocal of the denominator. ### $\frac {\frac {x^2 - 9}{3x}}{\frac {x + 3}{3x}} = \frac {x^2 - 9}{3x} \times \frac {3x}{x + 3} = \frac {x^2 - 9}{x + 3}$ ## Step 4: Factor and simplify The numerator, $x^2 - 9$, is a difference of squares and can be factored to $(x - 3)(x + 3)$. The denominator, $x + 3$, can be cancelled out with the same term in the numerator, leaving $x - 3$ as the simplified fraction. ### $\frac {x^2 - 9}{x + 3} = \frac {(x - 3)(x + 3)}{x + 3} = x - 3$

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Simplify the complex fraction. (frac (x)/(3)-(3)/(x))((1)/(3)+(1)/(x))

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