Finite Math Examples

$\frac{e^{14} - e^{- 14}}{e^{7} - e^{- 7}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $e^{14}$ as ${(e^{7})}^{2}$ .

$\frac{{(e^{7})}^{2} - e^{- 14}}{e^{7} - e^{- 7}}$

Step 1.2

Rewrite $e^{- 14}$ as ${(e^{- 7})}^{2}$ .

$\frac{{(e^{7})}^{2} - {(e^{- 7})}^{2}}{e^{7} - e^{- 7}}$

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 $a = e^{7}$ and $b = e^{- 7}$ .

$\frac{(e^{7} + e^{- 7}) (e^{7} - e^{- 7})}{e^{7} - e^{- 7}}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(e^{7} + \frac{1}{e^{7}}) (e^{7} - e^{- 7})}{e^{7} - e^{- 7}}$

Step 1.4.2

To write $e^{7}$ as a fraction with a common denominator, multiply by $\frac{e^{7}}{e^{7}}$ .

$\frac{(\frac{e^{7} e^{7}}{e^{7}} + \frac{1}{e^{7}}) (e^{7} - e^{- 7})}{e^{7} - e^{- 7}}$

Step 1.4.3

Combine the numerators over the common denominator.

$\frac{\frac{e^{7} e^{7} + 1}{e^{7}} (e^{7} - e^{- 7})}{e^{7} - e^{- 7}}$

Step 1.4.4

Multiply $e^{7}$ by $e^{7}$ by adding the exponents.

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Step 1.4.4.1

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

$\frac{\frac{e^{7 + 7} + 1}{e^{7}} (e^{7} - e^{- 7})}{e^{7} - e^{- 7}}$

Step 1.4.4.2

Add $7$ and $7$ .

$\frac{\frac{e^{14} + 1}{e^{7}} (e^{7} - e^{- 7})}{e^{7} - e^{- 7}}$

Step 1.4.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{e^{14} + 1}{e^{7}} (e^{7} - \frac{1}{e^{7}})}{e^{7} - e^{- 7}}$

Step 1.4.6

To write $e^{7}$ as a fraction with a common denominator, multiply by $\frac{e^{7}}{e^{7}}$ .

$\frac{\frac{e^{14} + 1}{e^{7}} (\frac{e^{7} e^{7}}{e^{7}} - \frac{1}{e^{7}})}{e^{7} - e^{- 7}}$

Step 1.4.7

Combine the numerators over the common denominator.

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{e^{7} e^{7} - 1}{e^{7}}}{e^{7} - e^{- 7}}$

Step 1.4.8

Rewrite $\frac{e^{7} e^{7} - 1}{e^{7}}$ in a factored form.

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Step 1.4.8.1

Multiply $e^{7}$ by $e^{7}$ by adding the exponents.

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Step 1.4.8.1.1

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

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{e^{7 + 7} - 1}{e^{7}}}{e^{7} - e^{- 7}}$

Step 1.4.8.1.2

Add $7$ and $7$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{e^{14} - 1}{e^{7}}}{e^{7} - e^{- 7}}$

Step 1.4.8.2

Rewrite $e^{14} - 1$ in a factored form.

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Step 1.4.8.2.1

Rewrite $e^{14}$ as ${(e^{7})}^{2}$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{{(e^{7})}^{2} - 1}{e^{7}}}{e^{7} - e^{- 7}}$

Step 1.4.8.2.2

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

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{{(e^{7})}^{2} - 1^{2}}{e^{7}}}{e^{7} - e^{- 7}}$

Step 1.4.8.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{7}$ and $b = 1$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{e^{7} - e^{- 7}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{e^{7} - \frac{1}{e^{7}}}$

Step 2.2

To write $e^{7}$ as a fraction with a common denominator, multiply by $\frac{e^{7}}{e^{7}}$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{e^{7} e^{7}}{e^{7}} - \frac{1}{e^{7}}}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{e^{7} e^{7} - 1}{e^{7}}}$

Step 2.4

Rewrite $\frac{e^{7} e^{7} - 1}{e^{7}}$ in a factored form.

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Step 2.4.1

Multiply $e^{7}$ by $e^{7}$ by adding the exponents.

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Step 2.4.1.1

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

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{e^{7 + 7} - 1}{e^{7}}}$

Step 2.4.1.2

Add $7$ and $7$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{e^{14} - 1}{e^{7}}}$

Step 2.4.2

Rewrite $e^{14} - 1$ in a factored form.

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Step 2.4.2.1

Rewrite $e^{14}$ as ${(e^{7})}^{2}$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{{(e^{7})}^{2} - 1}{e^{7}}}$

Step 2.4.2.2

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

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{{(e^{7})}^{2} - 1^{2}}{e^{7}}}$

Step 2.4.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{7}$ and $b = 1$ .

$\frac{\frac{e^{14} + 1}{e^{7}} \cdot \frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}{\frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}$

Step 3

Multiply $\frac{e^{14} + 1}{e^{7}}$ by $\frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}$ .

$\frac{\frac{(e^{14} + 1) ((e^{7} + 1) (e^{7} - 1))}{e^{7} e^{7}}}{\frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}$

Step 4

Multiply $e^{7}$ by $e^{7}$ by adding the exponents.

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Step 4.1

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

$\frac{\frac{(e^{14} + 1) (e^{7} + 1) (e^{7} - 1)}{e^{7 + 7}}}{\frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}$

Step 4.2

Add $7$ and $7$ .

$\frac{\frac{(e^{14} + 1) (e^{7} + 1) (e^{7} - 1)}{e^{14}}}{\frac{(e^{7} + 1) (e^{7} - 1)}{e^{7}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{(e^{14} + 1) (e^{7} + 1) (e^{7} - 1)}{e^{14}} \cdot \frac{e^{7}}{(e^{7} + 1) (e^{7} - 1)}$

Step 6

Combine.

$\frac{(e^{14} + 1) (e^{7} + 1) (e^{7} - 1) e^{7}}{e^{14} ((e^{7} + 1) (e^{7} - 1))}$

Step 7

Cancel the common factor of $e^{7} + 1$ .

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Step 7.1

Cancel the common factor.

$\frac{(e^{14} + 1) (e^{7} + 1) (e^{7} - 1) e^{7}}{e^{14} ((e^{7} + 1) (e^{7} - 1))}$

Step 7.2

Rewrite the expression.

$\frac{((e^{14} + 1) (e^{7} - 1)) e^{7}}{e^{14} (e^{7} - 1)}$

Step 8

Cancel the common factor of $e^{7} - 1$ .

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Step 8.1

Cancel the common factor.

$\frac{(e^{14} + 1) (e^{7} - 1) e^{7}}{e^{14} (e^{7} - 1)}$

Step 8.2

Rewrite the expression.

$\frac{(e^{14} + 1) e^{7}}{e^{14}}$

Step 9

Cancel the common factor of $e^{7}$ and $e^{14}$ .

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Step 9.1

Factor $e^{7}$ out of $(e^{14} + 1) e^{7}$ .

$\frac{e^{7} (e^{14} + 1)}{e^{14}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $e^{7}$ out of $e^{14}$ .

$\frac{e^{7} (e^{14} + 1)}{e^{7} e^{7}}$

Step 9.2.2

Cancel the common factor.

$\frac{e^{7} (e^{14} + 1)}{e^{7} e^{7}}$

Step 9.2.3

Rewrite the expression.

$\frac{e^{14} + 1}{e^{7}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{14} + 1}{e^{7}}$

Decimal Form:

$1096.63407031 \dots$