Algebra Examples

$\frac{x^{2} - 1}{6 (x + 9)} \div \frac{x + 1}{6 (x^{2} - 81)}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 1}{6 (x + 9)} \cdot \frac{6 (x^{2} - 81)}{x + 1}$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 2.1.1

Cancel the common factor.

$\frac{x^{2} - 1}{6 (x + 9)} \cdot \frac{6 (x^{2} - 81)}{x + 1}$

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Multiply $\frac{x^{2} - 1}{x + 9}$ by $\frac{x^{2} - 81}{x + 1}$ .

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

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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 $b = 1$ .

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

Step 3.3

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

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

Step 3.4

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 $b = 9$ .

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

Step 4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.1

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x + 9$ .

Tap for more steps...

Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide $(x - 1) (x - 9)$ by $1$ .

$(x - 1) (x - 9)$

Step 5

Expand $(x - 1) (x - 9)$ using the FOIL Method.

Tap for more steps...

Step 5.1

Apply the distributive property.

$x (x - 9) - 1 (x - 9)$

Step 5.2

Apply the distributive property.

$x \cdot x + x \cdot - 9 - 1 (x - 9)$

Step 5.3

Apply the distributive property.

$x \cdot x + x \cdot - 9 - 1 x - 1 \cdot - 9$

Step 6

Simplify and combine like terms.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $- 9$ to the left of $x$ .

$x^{2} - 9 \cdot x - 1 x - 1 \cdot - 9$

Step 6.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 9 x - x - 1 \cdot - 9$

Step 6.1.4

Multiply $- 1$ by $- 9$ .

$x^{2} - 9 x - x + 9$

Step 6.2

Subtract $x$ from $- 9 x$ .

$x^{2} - 10 x + 9$