Simplify (1/7+1/x)/(1/49-1/(x^2))

$\frac{\frac{1}{7} + \frac{1}{x}}{\frac{1}{49} - \frac{1}{x^{2}}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $49 x^{2}$ .

Multiply $\frac{\frac{1}{7} + \frac{1}{x}}{\frac{1}{49} - \frac{1}{x^{2}}}$ by $\frac{49 x^{2}}{49 x^{2}}$ .

$\frac{49 x^{2}}{49 x^{2}} \cdot \frac{\frac{1}{7} + \frac{1}{x}}{\frac{1}{49} - \frac{1}{x^{2}}}$

Combine.

$\frac{49 x^{2} (\frac{1}{7} + \frac{1}{x})}{49 x^{2} (\frac{1}{49} - \frac{1}{x^{2}})}$

Step 2

Apply the distributive property.

$\frac{49 x^{2} \frac{1}{7} + 49 x^{2} \frac{1}{x}}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $7$ .

Factor $7$ out of $49 x^{2}$ .

$\frac{7 (7 x^{2}) \frac{1}{7} + 49 x^{2} \frac{1}{x}}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor.

$\frac{7 (7 x^{2}) \frac{1}{7} + 49 x^{2} \frac{1}{x}}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Rewrite the expression.

$\frac{7 x^{2} + 49 x^{2} \frac{1}{x}}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor of $x$ .

Factor $x$ out of $49 x^{2}$ .

$\frac{7 x^{2} + x (49 x) \frac{1}{x}}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor.

$\frac{7 x^{2} + x (49 x) \frac{1}{x}}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Rewrite the expression.

$\frac{7 x^{2} + 49 x}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor of $49$ .

Factor $49$ out of $49 x^{2}$ .

$\frac{7 x^{2} + 49 x}{49 (x^{2}) \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor.

$\frac{7 x^{2} + 49 x}{49 x^{2} \frac{1}{49} + 49 x^{2} (- \frac{1}{x^{2}})}$

Rewrite the expression.

$\frac{7 x^{2} + 49 x}{x^{2} + 49 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor of $x^{2}$ .

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$\frac{7 x^{2} + 49 x}{x^{2} + 49 x^{2} \frac{- 1}{x^{2}}}$

Factor $x^{2}$ out of $49 x^{2}$ .

$\frac{7 x^{2} + 49 x}{x^{2} + x^{2} \cdot 49 \frac{- 1}{x^{2}}}$

Cancel the common factor.

$\frac{7 x^{2} + 49 x}{x^{2} + x^{2} \cdot 49 \frac{- 1}{x^{2}}}$

Rewrite the expression.

$\frac{7 x^{2} + 49 x}{x^{2} + 49 \cdot - 1}$

Multiply $49$ by $- 1$ .

$\frac{7 x^{2} + 49 x}{x^{2} - 49}$

Step 4

Factor $7 x$ out of $7 x^{2} + 49 x$ .

Factor $7 x$ out of $7 x^{2}$ .

$\frac{7 x (x) + 49 x}{x^{2} - 49}$

Factor $7 x$ out of $49 x$ .

$\frac{7 x (x) + 7 x (7)}{x^{2} - 49}$

Factor $7 x$ out of $7 x (x) + 7 x (7)$ .

$\frac{7 x (x + 7)}{x^{2} - 49}$

Step 5

Simplify the denominator.

Rewrite $49$ as $7^{2}$ .

$\frac{7 x (x + 7)}{x^{2} - 7^{2}}$

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 $b = 7$ .

$\frac{7 x (x + 7)}{(x + 7) (x - 7)}$

Step 6

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{7 x (x + 7)}{(x + 7) (x - 7)}$

Rewrite the expression.

$\frac{7 x}{x - 7}$

Simplify (1/7+1/x)/(1/49-1/(x^2))

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