Pre-Algebra Examples

$\frac{x^{4} - 1}{x^{4} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{{(x^{2})}^{2} - 1}{x^{4} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 1.2

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

$\frac{{(x^{2})}^{2} - 1^{2}}{x^{4} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 1.3

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

$\frac{(x^{2} + 1) (x^{2} - 1)}{x^{4} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 1.4

Simplify.

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Step 1.4.1

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

$\frac{(x^{2} + 1) (x^{2} - 1^{2})}{x^{4} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 1.4.2

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 $b = 1$ .

$\frac{(x^{2} + 1) (x + 1) (x - 1)}{x^{4} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{(x^{2} + 1) (x + 1) (x - 1)}{{(x^{2})}^{2} - 81} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 2.2

Rewrite $81$ as $9^{2}$ .

$\frac{(x^{2} + 1) (x + 1) (x - 1)}{{(x^{2})}^{2} - 9^{2}} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

$\frac{(x^{2} + 1) (x + 1) (x - 1)}{(x^{2} + 9) (x^{2} - 9)} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite $9$ as $3^{2}$ .

$\frac{(x^{2} + 1) (x + 1) (x - 1)}{(x^{2} + 9) (x^{2} - 3^{2})} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 2.4.2

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 $b = 3$ .

$\frac{(x^{2} + 1) (x + 1) (x - 1)}{(x^{2} + 9) (x + 3) (x - 3)} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

Factor $x^{2} + 1$ out of $(x^{2} + 1) (x + 1) (x - 1)$ .

$\frac{(x^{2} + 1) ((x + 1) (x - 1))}{(x^{2} + 9) (x + 3) (x - 3)} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 3.1.2

Cancel the common factor.

$\frac{(x^{2} + 1) ((x + 1) (x - 1))}{(x^{2} + 9) (x + 3) (x - 3)} \cdot \frac{x^{2} + 9}{x^{2} + 1}$

Step 3.1.3

Rewrite the expression.

$\frac{(x + 1) (x - 1)}{(x^{2} + 9) (x + 3) (x - 3)} \cdot (x^{2} + 9)$

Step 3.2

Cancel the common factor of $x^{2} + 9$ .

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Step 3.2.1

Factor $x^{2} + 9$ out of $(x^{2} + 9) (x + 3) (x - 3)$ .

$\frac{(x + 1) (x - 1)}{(x^{2} + 9) ((x + 3) (x - 3))} \cdot (x^{2} + 9)$

Step 3.2.2

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{(x^{2} + 9) ((x + 3) (x - 3))} \cdot (x^{2} + 9)$

Step 3.2.3

Rewrite the expression.

$\frac{(x + 1) (x - 1)}{(x + 3) (x - 3)}$