Algebra Examples

$\frac{16 x^{2}}{36 x^{2} - 1} \cdot \frac{42 x - 7}{14 x^{3}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $36 x^{2}$ as ${(6 x)}^{2}$ .

$\frac{16 x^{2}}{{(6 x)}^{2} - 1} \cdot \frac{42 x - 7}{14 x^{3}}$

Step 1.2

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

$\frac{16 x^{2}}{{(6 x)}^{2} - 1^{2}} \cdot \frac{42 x - 7}{14 x^{3}}$

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 $a = 6 x$ and $b = 1$ .

$\frac{16 x^{2}}{(6 x + 1) (6 x - 1)} \cdot \frac{42 x - 7}{14 x^{3}}$

Step 2

Simplify terms.

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Step 2.1

Factor $7$ out of $42 x - 7$ .

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Step 2.1.1

Factor $7$ out of $42 x$ .

$\frac{16 x^{2}}{(6 x + 1) (6 x - 1)} \cdot \frac{7 (6 x) - 7}{14 x^{3}}$

Step 2.1.2

Factor $7$ out of $- 7$ .

$\frac{16 x^{2}}{(6 x + 1) (6 x - 1)} \cdot \frac{7 (6 x) + 7 (- 1)}{14 x^{3}}$

Step 2.1.3

Factor $7$ out of $7 (6 x) + 7 (- 1)$ .

$\frac{16 x^{2}}{(6 x + 1) (6 x - 1)} \cdot \frac{7 (6 x - 1)}{14 x^{3}}$

Step 2.2

Combine.

$\frac{16 x^{2} (7 (6 x - 1))}{(6 x + 1) (6 x - 1) (14 x^{3})}$

Step 2.3

Cancel the common factor of $16$ and $14$ .

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Step 2.3.1

Factor $2$ out of $16 x^{2} (7 (6 x - 1))$ .

$\frac{2 (8 x^{2} (7 (6 x - 1)))}{(6 x + 1) (6 x - 1) (14 x^{3})}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $(6 x + 1) (6 x - 1) (14 x^{3})$ .

$\frac{2 (8 x^{2} (7 (6 x - 1)))}{2 ((6 x + 1) (6 x - 1) (7 x^{3}))}$

Step 2.3.2.2

Cancel the common factor.

$\frac{2 (8 x^{2} (7 (6 x - 1)))}{2 ((6 x + 1) (6 x - 1) (7 x^{3}))}$

Step 2.3.2.3

Rewrite the expression.

$\frac{8 x^{2} (7 (6 x - 1))}{(6 x + 1) (6 x - 1) (7 x^{3})}$

Step 2.4

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.4.1

Factor $x^{2}$ out of $8 x^{2} (7 (6 x - 1))$ .

$\frac{x^{2} (8 (7 (6 x - 1)))}{(6 x + 1) (6 x - 1) (7 x^{3})}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $x^{2}$ out of $(6 x + 1) (6 x - 1) (7 x^{3})$ .

$\frac{x^{2} (8 (7 (6 x - 1)))}{x^{2} ((6 x + 1) (6 x - 1) (7 x))}$

Step 2.4.2.2

Cancel the common factor.

$\frac{x^{2} (8 (7 (6 x - 1)))}{x^{2} ((6 x + 1) (6 x - 1) (7 x))}$

Step 2.4.2.3

Rewrite the expression.

$\frac{8 (7 (6 x - 1))}{(6 x + 1) (6 x - 1) (7 x)}$

Step 2.5

Cancel the common factor of $7$ .

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Step 2.5.1

Cancel the common factor.

$\frac{8 (7 (6 x - 1))}{(6 x + 1) (6 x - 1) (7 x)}$

Step 2.5.2

Rewrite the expression.

$\frac{8 (6 x - 1)}{(6 x + 1) (6 x - 1) (x)}$

Step 2.6

Cancel the common factor of $6 x - 1$ .

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Step 2.6.1

Cancel the common factor.

$\frac{8 (6 x - 1)}{(6 x + 1) (6 x - 1) x}$

Step 2.6.2

Rewrite the expression.

$\frac{8}{(6 x + 1) (x)}$

Step 2.7

Reorder factors in $\frac{8}{(6 x + 1) (x)}$ .

$\frac{8}{x (6 x + 1)}$