Finite Math Examples

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

Step 1

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

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Step 1.1

Multiply $\frac{36 - \frac{1}{x^{2}}}{6 - \frac{1}{x}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $x^{2} \cdot 36$ as ${(x \cdot 6)}^{2}$ .

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

Step 4.2

Rewrite $1$ as $1^{2}$ .

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x \cdot 6$ and $b = 1$ .

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

Step 4.4

Simplify.

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Step 4.4.1

Move $6$ to the left of $x$ .

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

Step 4.4.2

Move $6$ to the left of $x$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x$ out of $x^{2} \cdot 6 + x \cdot - 1$ .

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Step 5.1.1

Factor $x$ out of $x^{2} \cdot 6$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Move $6$ to the left of $x$ .

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

Step 6

Cancel the common factor of $6 x - 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{6 x + 1}{x}$