Simplify (1/x-x/9)/((x^2)/9+x/3)

$\frac{\frac{1}{x} - \frac{x}{9}}{\frac{x^{2}}{9} + \frac{x}{3}}$

Multiply the numerator and denominator of the complex fraction by $x \cdot 9$ .

Multiply $\frac{\frac{1}{x} - \frac{x}{9}}{\frac{x^{2}}{9} + \frac{x}{3}}$ by $\frac{x \cdot 9}{x \cdot 9}$ .

$\frac{x \cdot 9}{x \cdot 9} \cdot \frac{\frac{1}{x} - \frac{x}{9}}{\frac{x^{2}}{9} + \frac{x}{3}}$

Combine.

$\frac{x \cdot 9 (\frac{1}{x} - \frac{x}{9})}{x \cdot 9 (\frac{x^{2}}{9} + \frac{x}{3})}$

Apply the distributive property.

$\frac{x \cdot 9 \frac{1}{x} + x \cdot 9 (- \frac{x}{9})}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 9$ .

$\frac{x (9) \frac{1}{x} + x \cdot 9 (- \frac{x}{9})}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Cancel the common factor.

$\frac{x \cdot 9 \frac{1}{x} + x \cdot 9 (- \frac{x}{9})}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Rewrite the expression.

$\frac{9 + x \cdot 9 (- \frac{x}{9})}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$\frac{9 + x \cdot 9 \frac{- x}{9}}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Factor $9$ out of $x \cdot 9$ .

$\frac{9 + 9 \cdot x \frac{- x}{9}}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Cancel the common factor.

$\frac{9 + 9 \cdot x \frac{- x}{9}}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Rewrite the expression.

$\frac{9 + x (- x)}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Raise $x$ to the power of $1$ .

$\frac{9 - (x^{1} x)}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Raise $x$ to the power of $1$ .

$\frac{9 - (x^{1} x^{1})}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 - x^{1 + 1}}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Add $1$ and $1$ .

$\frac{9 - x^{2}}{x \cdot 9 \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Cancel the common factor of $9$ .

Factor $9$ out of $x \cdot 9$ .

$\frac{9 - x^{2}}{9 \cdot x \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Cancel the common factor.

$\frac{9 - x^{2}}{9 \cdot x \frac{x^{2}}{9} + x \cdot 9 \frac{x}{3}}$

Rewrite the expression.

$\frac{9 - x^{2}}{x \cdot x^{2} + x \cdot 9 \frac{x}{3}}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$\frac{9 - x^{2}}{x^{1} x^{2} + x \cdot 9 \frac{x}{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 - x^{2}}{x^{1 + 2} + x \cdot 9 \frac{x}{3}}$

Add $1$ and $2$ .

$\frac{9 - x^{2}}{x^{3} + x \cdot 9 \frac{x}{3}}$

Cancel the common factor of $3$ .

Factor $3$ out of $x \cdot 9$ .

$\frac{9 - x^{2}}{x^{3} + 3 (x \cdot 3) \frac{x}{3}}$

Cancel the common factor.

$\frac{9 - x^{2}}{x^{3} + 3 (x \cdot 3) \frac{x}{3}}$

Rewrite the expression.

$\frac{9 - x^{2}}{x^{3} + x \cdot 3 x}$

Raise $x$ to the power of $1$ .

$\frac{9 - x^{2}}{x^{3} + 3 (x^{1} x)}$

Raise $x$ to the power of $1$ .

$\frac{9 - x^{2}}{x^{3} + 3 (x^{1} x^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 - x^{2}}{x^{3} + 3 x^{1 + 1}}$

Add $1$ and $1$ .

$\frac{9 - x^{2}}{x^{3} + 3 x^{2}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - x^{2}}{x^{3} + 3 x^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{(3 + x) (3 - x)}{x^{3} + 3 x^{2}}$

Simplify terms.

Factor $x^{2}$ out of $x^{3} + 3 x^{2}$ .

Factor $x^{2}$ out of $x^{3}$ .

$\frac{(3 + x) (3 - x)}{x^{2} x + 3 x^{2}}$

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{(3 + x) (3 - x)}{x^{2} x + x^{2} \cdot 3}$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

$\frac{(3 + x) (3 - x)}{x^{2} (x + 3)}$

Cancel the common factor of $3 + x$ and $x + 3$ .

Reorder terms.

$\frac{(x + 3) (3 - x)}{x^{2} (x + 3)}$

Cancel the common factor.

$\frac{(x + 3) (3 - x)}{x^{2} (x + 3)}$

Rewrite the expression.

$\frac{3 - x}{x^{2}}$

Simplify (1/x-x/9)/((x^2)/9+x/3)

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