Basic Math Examples

$\frac{2 a^{2} - 2}{10 a^{2} - 160} \cdot \frac{50 a + 200}{5 a - 5}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $2 a^{2} - 2$ .

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Step 1.1.1

Factor $2$ out of $2 a^{2}$ .

$\frac{2 (a^{2}) - 2}{10 a^{2} - 160} \cdot \frac{50 a + 200}{5 a - 5}$

Step 1.1.2

Factor $2$ out of $- 2$ .

$\frac{2 (a^{2}) + 2 (- 1)}{10 a^{2} - 160} \cdot \frac{50 a + 200}{5 a - 5}$

Step 1.1.3

Factor $2$ out of $2 (a^{2}) + 2 (- 1)$ .

$\frac{2 (a^{2} - 1)}{10 a^{2} - 160} \cdot \frac{50 a + 200}{5 a - 5}$

Step 1.2

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

$\frac{2 (a^{2} - 1^{2})}{10 a^{2} - 160} \cdot \frac{50 a + 200}{5 a - 5}$

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 $a = a$ and $b = 1$ .

$\frac{2 (a + 1) (a - 1)}{10 a^{2} - 160} \cdot \frac{50 a + 200}{5 a - 5}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $10$ out of $10 a^{2} - 160$ .

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Step 2.1.1

Factor $10$ out of $10 a^{2}$ .

$\frac{2 (a + 1) (a - 1)}{10 (a^{2}) - 160} \cdot \frac{50 a + 200}{5 a - 5}$

Step 2.1.2

Factor $10$ out of $- 160$ .

$\frac{2 (a + 1) (a - 1)}{10 a^{2} + 10 \cdot - 16} \cdot \frac{50 a + 200}{5 a - 5}$

Step 2.1.3

Factor $10$ out of $10 a^{2} + 10 \cdot - 16$ .

$\frac{2 (a + 1) (a - 1)}{10 (a^{2} - 16)} \cdot \frac{50 a + 200}{5 a - 5}$

Step 2.2

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

$\frac{2 (a + 1) (a - 1)}{10 (a^{2} - 4^{2})} \cdot \frac{50 a + 200}{5 a - 5}$

Step 2.3

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

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 a + 200}{5 a - 5}$

Step 3

Simplify terms.

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Step 3.1

Factor $50$ out of $50 a + 200$ .

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Step 3.1.1

Factor $50$ out of $50 a$ .

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 (a) + 200}{5 a - 5}$

Step 3.1.2

Factor $50$ out of $200$ .

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 a + 50 \cdot 4}{5 a - 5}$

Step 3.1.3

Factor $50$ out of $50 a + 50 \cdot 4$ .

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{5 a - 5}$

Step 3.2

Factor $5$ out of $5 a - 5$ .

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Step 3.2.1

Factor $5$ out of $5 a$ .

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{5 (a) - 5}$

Step 3.2.2

Factor $5$ out of $- 5$ .

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{5 (a) + 5 (- 1)}$

Step 3.2.3

Factor $5$ out of $5 (a) + 5 (- 1)$ .

$\frac{2 (a + 1) (a - 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{5 (a - 1)}$

Step 3.3

Cancel the common factor of $a - 1$ .

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Step 3.3.1

Factor $a - 1$ out of $2 (a + 1) (a - 1)$ .

$\frac{(a - 1) (2 (a + 1))}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{5 (a - 1)}$

Step 3.3.2

Factor $a - 1$ out of $5 (a - 1)$ .

$\frac{(a - 1) (2 (a + 1))}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{(a - 1) \cdot 5}$

Step 3.3.3

Cancel the common factor.

$\frac{(a - 1) (2 (a + 1))}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{(a - 1) \cdot 5}$

Step 3.3.4

Rewrite the expression.

$\frac{2 (a + 1)}{10 (a + 4) (a - 4)} \cdot \frac{50 (a + 4)}{5}$

Step 3.4

Cancel the common factor of $10 (a + 4)$ .

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Step 3.4.1

Factor $10 (a + 4)$ out of $50 (a + 4)$ .

$\frac{2 (a + 1)}{10 (a + 4) (a - 4)} \cdot \frac{10 (a + 4) (5)}{5}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (a + 1)}{10 (a + 4) (a - 4)} \cdot \frac{10 (a + 4) \cdot 5}{5}$

Step 3.4.3

Rewrite the expression.

$\frac{2 (a + 1)}{a - 4} \cdot \frac{5}{5}$

Step 3.5

Multiply $\frac{2 (a + 1)}{a - 4}$ by $\frac{5}{5}$ .

$\frac{2 (a + 1) \cdot 5}{(a - 4) \cdot 5}$

Step 3.6

Multiply $5$ by $2$ .

$\frac{10 (a + 1)}{(a - 4) \cdot 5}$

Step 3.7

Cancel the common factor of $10$ and $5$ .

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Step 3.7.1

Factor $5$ out of $10 (a + 1)$ .

$\frac{5 (2 (a + 1))}{(a - 4) \cdot 5}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $5$ out of $(a - 4) \cdot 5$ .

$\frac{5 (2 (a + 1))}{5 \cdot (a - 4)}$

Step 3.7.2.2

Cancel the common factor.

$\frac{5 (2 (a + 1))}{5 \cdot (a - 4)}$

Step 3.7.2.3

Rewrite the expression.

$\frac{2 (a + 1)}{a - 4}$