Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

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 $a = z^{2}$ and $b = 9$ .

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

Step 1.4

Simplify.

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Step 1.4.1

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

$\frac{(z^{2} + 9) (z^{2} - 3^{2})}{z^{4} - 1} \cdot \frac{z^{2} + 1}{z^{2} + 9}$

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 $a = z$ and $b = 3$ .

$\frac{(z^{2} + 9) (z + 3) (z - 3)}{z^{4} - 1} \cdot \frac{z^{2} + 1}{z^{2} + 9}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

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 $a = z^{2}$ and $b = 1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

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 $a = z$ and $b = 1$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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