Finite Math Examples

$\frac{2 d^{2} - 2}{8 d^{2} - 72} \cdot \frac{32 d + 96}{4 d - 4}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

$\frac{2 (d^{2}) - 2}{8 d^{2} - 72} \cdot \frac{32 d + 96}{4 d - 4}$

Step 1.1.2

Factor $2$ out of $- 2$ .

$\frac{2 (d^{2}) + 2 (- 1)}{8 d^{2} - 72} \cdot \frac{32 d + 96}{4 d - 4}$

Step 1.1.3

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

$\frac{2 (d^{2} - 1)}{8 d^{2} - 72} \cdot \frac{32 d + 96}{4 d - 4}$

Step 1.2

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

$\frac{2 (d^{2} - 1^{2})}{8 d^{2} - 72} \cdot \frac{32 d + 96}{4 d - 4}$

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

$\frac{2 (d + 1) (d - 1)}{8 d^{2} - 72} \cdot \frac{32 d + 96}{4 d - 4}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $8$ out of $8 d^{2} - 72$ .

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Step 2.1.1

Factor $8$ out of $8 d^{2}$ .

$\frac{2 (d + 1) (d - 1)}{8 (d^{2}) - 72} \cdot \frac{32 d + 96}{4 d - 4}$

Step 2.1.2

Factor $8$ out of $- 72$ .

$\frac{2 (d + 1) (d - 1)}{8 d^{2} + 8 \cdot - 9} \cdot \frac{32 d + 96}{4 d - 4}$

Step 2.1.3

Factor $8$ out of $8 d^{2} + 8 \cdot - 9$ .

$\frac{2 (d + 1) (d - 1)}{8 (d^{2} - 9)} \cdot \frac{32 d + 96}{4 d - 4}$

Step 2.2

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

$\frac{2 (d + 1) (d - 1)}{8 (d^{2} - 3^{2})} \cdot \frac{32 d + 96}{4 d - 4}$

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 $a = d$ and $b = 3$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 d + 96}{4 d - 4}$

Step 3

Simplify terms.

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Step 3.1

Factor $32$ out of $32 d + 96$ .

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Step 3.1.1

Factor $32$ out of $32 d$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d) + 96}{4 d - 4}$

Step 3.1.2

Factor $32$ out of $96$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 d + 32 \cdot 3}{4 d - 4}$

Step 3.1.3

Factor $32$ out of $32 d + 32 \cdot 3$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{4 d - 4}$

Step 3.2

Factor $4$ out of $4 d - 4$ .

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Step 3.2.1

Factor $4$ out of $4 d$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{4 (d) - 4}$

Step 3.2.2

Factor $4$ out of $- 4$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{4 (d) + 4 (- 1)}$

Step 3.2.3

Factor $4$ out of $4 (d) + 4 (- 1)$ .

$\frac{2 (d + 1) (d - 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{4 (d - 1)}$

Step 3.3

Cancel the common factor of $2 (d - 1)$ .

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Step 3.3.1

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

$\frac{2 (d - 1) (d + 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{4 (d - 1)}$

Step 3.3.2

Factor $2 (d - 1)$ out of $4 (d - 1)$ .

$\frac{2 (d - 1) (d + 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{2 (d - 1) (2)}$

Step 3.3.3

Cancel the common factor.

$\frac{2 (d - 1) (d + 1)}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{2 (d - 1) \cdot 2}$

Step 3.3.4

Rewrite the expression.

$\frac{d + 1}{8 (d + 3) (d - 3)} \cdot \frac{32 (d + 3)}{2}$

Step 3.4

Cancel the common factor of $8 (d + 3)$ .

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Step 3.4.1

Factor $8 (d + 3)$ out of $32 (d + 3)$ .

$\frac{d + 1}{8 (d + 3) (d - 3)} \cdot \frac{8 (d + 3) (4)}{2}$

Step 3.4.2

Cancel the common factor.

$\frac{d + 1}{8 (d + 3) (d - 3)} \cdot \frac{8 (d + 3) \cdot 4}{2}$

Step 3.4.3

Rewrite the expression.

$\frac{d + 1}{d - 3} \cdot \frac{4}{2}$

Step 3.5

Multiply $\frac{d + 1}{d - 3}$ by $\frac{4}{2}$ .

$\frac{(d + 1) \cdot 4}{(d - 3) \cdot 2}$

Step 3.6

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

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Step 3.6.1

Factor $2$ out of $(d + 1) \cdot 4$ .

$\frac{2 ((d + 1) \cdot 2)}{(d - 3) \cdot 2}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $2$ out of $(d - 3) \cdot 2$ .

$\frac{2 ((d + 1) \cdot 2)}{2 \cdot (d - 3)}$

Step 3.6.2.2

Cancel the common factor.

$\frac{2 ((d + 1) \cdot 2)}{2 \cdot (d - 3)}$

Step 3.6.2.3

Rewrite the expression.

$\frac{(d + 1) \cdot 2}{d - 3}$

Step 3.7

Move $2$ to the left of $d + 1$ .

$\frac{2 (d + 1)}{d - 3}$