Trigonometry Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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 $b = 4$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

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 $b = 4$ .

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of $4$ and $2$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $(x + 4) (x - 4) (2 x)$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $x$ out of $(x + 4) (x - 4) (x)$ .

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x - 4$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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