Pre-Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $49$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Multiply $49$ by $- 1$ .

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

Step 3.4

Cancel the common factor of $7$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

Cancel the common factor of $x$ .

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Step 3.5.1

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

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

Step 3.5.2

Cancel the common factor.

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

Step 3.5.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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 $b = 7$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Cancel the common factor of $x + 7$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Split the fraction $\frac{x - 7}{7 x}$ into two fractions.

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

Step 8

Cancel the common factor of $x$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$\frac{1}{7} + \frac{- 7}{7 x}$

Step 9

Cancel the common factor of $- 7$ and $7$ .

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Step 9.1

Factor $7$ out of $- 7$ .

$\frac{1}{7} + \frac{7 \cdot - 1}{7 x}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $7$ out of $7 x$ .

$\frac{1}{7} + \frac{7 \cdot - 1}{7 (x)}$

Step 9.2.2

Cancel the common factor.

$\frac{1}{7} + \frac{7 \cdot - 1}{7 x}$

Step 9.2.3

Rewrite the expression.

$\frac{1}{7} + \frac{- 1}{x}$

Step 10

Move the negative in front of the fraction.

$\frac{1}{7} - \frac{1}{x}$