Pre-Algebra Examples

$\frac{\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}}}{\frac{x^{2 - 11 x + 28}}{x^{2 - x - 42}}}$

Step 1

Cancel the common factor of $x^{2 - 11 x + 28}$ and $x^{2 - x - 42}$ .

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Step 1.1

Factor $x^{2 - x - 42}$ out of $x^{2 - 11 x + 28}$ .

$\frac{\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}}}{\frac{x^{2 - x - 42} x^{- 11 x + 30 - (2 - x - 42)}}{x^{2 - x - 42}}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

$\frac{\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}}}{\frac{x^{2 - x - 42} x^{- 11 x + 30 - (2 - x - 42)}}{x^{2 - x - 42} \cdot 1}}$

Step 1.2.2

Cancel the common factor.

$\frac{\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}}}{\frac{x^{2 - x - 42} x^{- 11 x + 30 - (2 - x - 42)}}{x^{2 - x - 42} \cdot 1}}$

Step 1.2.3

Rewrite the expression.

$\frac{\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}}}{\frac{x^{- 11 x + 30 - (2 - x - 42)}}{1}}$

Step 1.2.4

Divide $x^{- 11 x + 30 - (2 - x - 42)}$ by $1$ .

$\frac{\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}}}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 + 2 x - x^{2}}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $8$ as $9$ plus $- 1$

$\frac{9 - 1 + 2 x - x^{2}}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

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

$\frac{9 - (1^{2} - 2 x + x^{2})}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot 1 \cdot x$

Step 3.2.3

Rewrite the polynomial.

$\frac{9 - (1^{2} - 2 \cdot 1 \cdot x + x^{2})}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = x$ .

$\frac{9 - {(1 - x)}^{2}}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.3

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - {(1 - x)}^{2}}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

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 $a = 3$ and $b = 1 - x$ .

$\frac{(3 + 1 - x) (3 - (1 - x))}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.5

Simplify.

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Step 3.5.1

Add $3$ and $1$ .

$\frac{(4 - x) (3 - (1 - x))}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.5.2

Apply the distributive property.

$\frac{(4 - x) (3 - 1 \cdot 1 - - x)}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.5.3

Multiply $- 1$ by $1$ .

$\frac{(4 - x) (3 - 1 - - x)}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.5.4

Multiply $- - x$ .

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Step 3.5.4.1

Multiply $- 1$ by $- 1$ .

$\frac{(4 - x) (3 - 1 + 1 x)}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.5.4.2

Multiply $x$ by $1$ .

$\frac{(4 - x) (3 - 1 + x)}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 3.5.5

Subtract $1$ from $3$ .

$\frac{(4 - x) (2 + x)}{x^{2 + 7 x + 10}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 4

Add $2$ and $10$ .

$\frac{(4 - x) (2 + x)}{x^{7 x + 12}} \cdot \frac{1}{x^{- 11 x + 30 - (2 - x - 42)}}$

Step 5

Combine.

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12} x^{- 11 x + 30 - (2 - x - 42)}}$

Step 6

Multiply $x^{7 x + 12}$ by $x^{- 11 x + 30 - (2 - x - 42)}$ by adding the exponents.

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Step 6.1

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

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 11 x + 30 - (2 - x - 42)}}$

Step 6.2

Subtract $42$ from $2$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 11 x + 30 - (- x - 40)}}$

Step 6.3

Simplify each term.

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Step 6.3.1

Apply the distributive property.

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 11 x + 30 - - x - - 40}}$

Step 6.3.2

Multiply $- - x$ .

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Step 6.3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 11 x + 30 + 1 x - - 40}}$

Step 6.3.2.2

Multiply $x$ by $1$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 11 x + 30 + x - - 40}}$

Step 6.3.3

Multiply $- 1$ by $- 40$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 11 x + 30 + x + 40}}$

Step 6.4

Add $- 11 x$ and $x$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 10 x + 30 + 40}}$

Step 6.5

Add $30$ and $40$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{7 x + 12 - 10 x + 70}}$

Step 6.6

Subtract $10 x$ from $7 x$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{- 3 x + 12 + 70}}$

Step 6.7

Add $12$ and $70$ .

$\frac{(4 - x) (2 + x) \cdot 1}{x^{- 3 x + 82}}$

Step 7

Multiply $(4 - x) (2 + x)$ by $1$ .

$\frac{(4 - x) (2 + x)}{x^{- 3 x + 82}}$

Step 8

Expand $(4 - x) (2 + x)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$\frac{4 (2 + x) - x (2 + x)}{x^{- 3 x + 82}}$

Step 8.2

Apply the distributive property.

$\frac{4 \cdot 2 + 4 x - x (2 + x)}{x^{- 3 x + 82}}$

Step 8.3

Apply the distributive property.

$\frac{4 \cdot 2 + 4 x - x \cdot 2 - x \cdot x}{x^{- 3 x + 82}}$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $4$ by $2$ .

$\frac{8 + 4 x - x \cdot 2 - x \cdot x}{x^{- 3 x + 82}}$

Step 9.1.2

Multiply $2$ by $- 1$ .

$\frac{8 + 4 x - 2 x - x \cdot x}{x^{- 3 x + 82}}$

Step 9.1.3

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

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Step 9.1.3.1

Move $x$ .

$\frac{8 + 4 x - 2 x - (x \cdot x)}{x^{- 3 x + 82}}$

Step 9.1.3.2

Multiply $x$ by $x$ .

$\frac{8 + 4 x - 2 x - x^{2}}{x^{- 3 x + 82}}$

Step 9.2

Subtract $2 x$ from $4 x$ .

$\frac{8 + 2 x - x^{2}}{x^{- 3 x + 82}}$

Step 10

Split the fraction $\frac{8 + 2 x - x^{2}}{x^{- 3 x + 82}}$ into two fractions.

$\frac{8 + 2 x}{x^{- 3 x + 82}} + \frac{- x^{2}}{x^{- 3 x + 82}}$

Step 11

Split the fraction $\frac{8 + 2 x}{x^{- 3 x + 82}}$ into two fractions.

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} + \frac{- x^{2}}{x^{- 3 x + 82}}$

Step 12

Cancel the common factor of $x^{2}$ and $x^{- 3 x + 82}$ .

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Step 12.1

Factor $x^{- 3 x + 82}$ out of $- x^{2}$ .

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} + \frac{x^{- 3 x + 82} (- x^{2 - (- 3 x + 82)})}{x^{- 3 x + 82}}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Multiply by $1$ .

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} + \frac{x^{- 3 x + 82} (- x^{2 - (- 3 x + 82)})}{x^{- 3 x + 82} \cdot 1}$

Step 12.2.2

Cancel the common factor.

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} + \frac{x^{- 3 x + 82} (- x^{2 - (- 3 x + 82)})}{x^{- 3 x + 82} \cdot 1}$

Step 12.2.3

Rewrite the expression.

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} + \frac{- x^{2 - (- 3 x + 82)}}{1}$

Step 12.2.4

Divide $- x^{2 - (- 3 x + 82)}$ by $1$ .

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} - x^{2 - (- 3 x + 82)}$

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} - x^{2 - (- 3 x) - 1 \cdot 82}$

Step 13.2

Multiply $- 3$ by $- 1$ .

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} - x^{2 + 3 x - 1 \cdot 82}$

Step 13.3

Multiply $- 1$ by $82$ .

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} - x^{2 + 3 x - 82}$

Step 14

Subtract $82$ from $2$ .

$\frac{8}{x^{- 3 x + 82}} + \frac{2 x}{x^{- 3 x + 82}} - x^{3 x - 80}$