Pre-Algebra Examples

$\frac{x^{3} - 64}{x^{3} + 64} \div \frac{x^{2} - 16}{x^{2} - 4 x + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{3} - 64}{x^{3} + 64} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $64$ as $4^{3}$ .

$\frac{x^{3} - 4^{3}}{x^{3} + 64} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

$\frac{(x - 4) (x^{2} + x \cdot 4 + 4^{2})}{x^{3} + 64} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Move $4$ to the left of $x$ .

$\frac{(x - 4) (x^{2} + 4 \cdot x + 4^{2})}{x^{3} + 64} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 2.3.2

Raise $4$ to the power of $2$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{x^{3} + 64} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Rewrite $64$ as $4^{3}$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{x^{3} + 4^{3}} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 3.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 4$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x + 4) (x^{2} - x \cdot 4 + 4^{2})} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Multiply $4$ by $- 1$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x + 4) (x^{2} - 4 x + 4^{2})} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 3.3.2

Raise $4$ to the power of $2$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x + 4) (x^{2} - 4 x + 16)} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Cancel the common factor of $x^{2} - 4 x + 16$ .

Tap for more steps...

Step 4.1.1

Factor $x^{2} - 4 x + 16$ out of $(x + 4) (x^{2} - 4 x + 16)$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x^{2} - 4 x + 16) (x + 4)} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 4.1.2

Cancel the common factor.

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x^{2} - 4 x + 16) (x + 4)} \cdot \frac{x^{2} - 4 x + 16}{x^{2} - 16}$

Step 4.1.3

Rewrite the expression.

$\frac{(x - 4) (x^{2} + 4 x + 16)}{x + 4} \cdot \frac{1}{x^{2} - 16}$

Step 4.2

Multiply $\frac{(x - 4) (x^{2} + 4 x + 16)}{x + 4}$ by $\frac{1}{x^{2} - 16}$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x + 4) (x^{2} - 16)}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Rewrite $16$ as $4^{2}$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x + 4) (x^{2} - 4^{2})}$

Step 5.2

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 $b = 4$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{(x + 4) (x + 4) (x - 4)}$

Step 5.3

Combine exponents.

Tap for more steps...

Step 5.3.1

Raise $x + 4$ to the power of $1$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{{(x + 4)}^{1} (x + 4) (x - 4)}$

Step 5.3.2

Raise $x + 4$ to the power of $1$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{{(x + 4)}^{1} {(x + 4)}^{1} (x - 4)}$

Step 5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(x - 4) (x^{2} + 4 x + 16)}{{(x + 4)}^{1 + 1} (x - 4)}$

Step 5.3.4

Add $1$ and $1$ .

$\frac{(x - 4) (x^{2} + 4 x + 16)}{{(x + 4)}^{2} (x - 4)}$

Step 6

Cancel the common factor of $x - 4$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\frac{(x - 4) (x^{2} + 4 x + 16)}{{(x + 4)}^{2} (x - 4)}$

Step 6.2

Rewrite the expression.

$\frac{x^{2} + 4 x + 16}{{(x + 4)}^{2}}$