Simplify (x/36-1/x)/(1-6/x)

$\frac{\frac{x}{36} - \frac{1}{x}}{1 - \frac{6}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $36 x$ .

Multiply $\frac{\frac{x}{36} - \frac{1}{x}}{1 - \frac{6}{x}}$ by $\frac{36 x}{36 x}$ .

$\frac{36 x}{36 x} \cdot \frac{\frac{x}{36} - \frac{1}{x}}{1 - \frac{6}{x}}$

Combine.

$\frac{36 x (\frac{x}{36} - \frac{1}{x})}{36 x (1 - \frac{6}{x})}$

Step 2

Apply the distributive property.

$\frac{36 x \frac{x}{36} + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $36$ .

Factor $36$ out of $36 x$ .

$\frac{36 (x) \frac{x}{36} + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Cancel the common factor.

$\frac{36 x \frac{x}{36} + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Rewrite the expression.

$\frac{x \cdot x + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

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

$\frac{x^{1} x + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

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

$\frac{x^{1} x^{1} + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

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

$\frac{x^{1 + 1} + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Add $1$ and $1$ .

$\frac{x^{2} + 36 x (- \frac{1}{x})}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x^{2} + 36 x \frac{- 1}{x}}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Factor $x$ out of $36 x$ .

$\frac{x^{2} + x \cdot 36 \frac{- 1}{x}}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Cancel the common factor.

$\frac{x^{2} + x \cdot 36 \frac{- 1}{x}}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Rewrite the expression.

$\frac{x^{2} + 36 \cdot - 1}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Multiply $36$ by $- 1$ .

$\frac{x^{2} - 36}{36 x \cdot 1 + 36 x (- \frac{6}{x})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{6}{x}$ into the numerator.

$\frac{x^{2} - 36}{36 x \cdot 1 + 36 x \frac{- 6}{x}}$

Factor $x$ out of $36 x$ .

$\frac{x^{2} - 36}{36 x \cdot 1 + x \cdot 36 \frac{- 6}{x}}$

Cancel the common factor.

$\frac{x^{2} - 36}{36 x \cdot 1 + x \cdot 36 \frac{- 6}{x}}$

Rewrite the expression.

$\frac{x^{2} - 36}{36 x \cdot 1 + 36 \cdot - 6}$

Multiply $36$ by $- 6$ .

$\frac{x^{2} - 36}{36 x \cdot 1 - 216}$

Step 4

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} - 6^{2}}{36 x \cdot 1 - 216}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\frac{(x + 6) (x - 6)}{36 x \cdot 1 - 216}$

Step 5

Simplify the denominator.

Factor $36$ out of $36 x \cdot 1 - 216$ .

Factor $36$ out of $36 x \cdot 1$ .

$\frac{(x + 6) (x - 6)}{36 (x \cdot 1) - 216}$

Factor $36$ out of $- 216$ .

$\frac{(x + 6) (x - 6)}{36 (x \cdot 1) + 36 \cdot - 6}$

Factor $36$ out of $36 (x \cdot 1) + 36 \cdot - 6$ .

$\frac{(x + 6) (x - 6)}{36 (x \cdot 1 - 6)}$

Multiply $x$ by $1$ .

$\frac{(x + 6) (x - 6)}{36 (x - 6)}$

Step 6

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{(x + 6) (x - 6)}{36 (x - 6)}$

Rewrite the expression.

$\frac{x + 6}{36}$

Simplify (x/36-1/x)/(1-6/x)

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