Pre-Algebra Examples

$\frac{x^{2} - 49}{7 y} \div \frac{7 - x}{21 x y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 49}{7 y} \cdot \frac{21 x y}{7 - x}$

Step 2

Cancel the common factor of $7 y$ .

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Step 2.1

Factor $7 y$ out of $21 x y$ .

$\frac{x^{2} - 49}{7 y} \cdot \frac{7 y (3 x)}{7 - x}$

Step 2.2

Cancel the common factor.

$\frac{x^{2} - 49}{7 y} \cdot \frac{7 y (3 x)}{7 - x}$

Step 2.3

Rewrite the expression.

$(x^{2} - 49) \frac{3 x}{7 - x}$

Step 3

Multiply $x^{2} - 49$ by $\frac{3 x}{7 - x}$ .

$\frac{(x^{2} - 49) (3 x)}{7 - x}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $49$ as $7^{2}$ .

$\frac{(x^{2} - 7^{2}) \cdot 3 x}{7 - x}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{(x + 7) (x - 7) \cdot 3 x}{7 - x}$

Step 5

Cancel the common factor of $x - 7$ and $7 - x$ .

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Step 5.1

Factor $- 1$ out of $x$ .

$\frac{(x + 7) (- 1 (- x) - 7) \cdot 3 x}{7 - x}$

Step 5.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{(x + 7) (- 1 (- x) - 1 (7)) \cdot 3 x}{7 - x}$

Step 5.3

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

$\frac{(x + 7) (- 1 (- x + 7)) \cdot 3 x}{7 - x}$

Step 5.4

Reorder terms.

$\frac{(x + 7) (- 1 (- x + 7)) \cdot 3 x}{- x + 7}$

Step 5.5

Cancel the common factor.

$\frac{(x + 7) (- 1 (- x + 7)) \cdot 3 x}{- x + 7}$

Step 5.6

Divide $((x + 7) \cdot (- 1)) \cdot 3 x$ by $1$ .

$((x + 7) \cdot (- 1)) \cdot 3 x$

Step 6

Apply the distributive property.

$(x \cdot - 1 + 7 \cdot - 1) \cdot 3 x$

Step 7

Move $- 1$ to the left of $x$ .

$(- 1 \cdot x + 7 \cdot - 1) \cdot 3 x$

Step 8

Multiply $7$ by $- 1$ .

$(- 1 \cdot x - 7) \cdot 3 x$

Step 9

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

$(- x - 7) \cdot 3 x$

Step 10

Apply the distributive property.

$(- x \cdot 3 - 7 \cdot 3) x$

Step 11

Multiply $3$ by $- 1$ .

$(- 3 x - 7 \cdot 3) x$

Step 12

Multiply $- 7$ by $3$ .

$(- 3 x - 21) x$

Step 13

Apply the distributive property.

$- 3 x \cdot x - 21 x$

Step 14

Multiply $x$ by $x$ by adding the exponents.

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Step 14.1

Move $x$ .

$- 3 (x \cdot x) - 21 x$

Step 14.2

Multiply $x$ by $x$ .

$- 3 x^{2} - 21 x$