Finite Math Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

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 $a = 1$ and $b = x \cdot x$ .

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

Step 4.4

Simplify.

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Step 4.4.1

Multiply $x$ by $x$ .

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

Step 4.4.2

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

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

Step 4.4.3

Rewrite $x \cdot x$ as $x^{2}$ .

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

Step 4.4.4

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

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

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Factor $x$ out of $x^{1}$ .

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.2

Multiply $x$ by $x$ .

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

Step 6

Cancel the common factor of $1 + x^{2}$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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