Calculus Examples

$((x e^{- 8} - e^{- 8}) - (x e^{0} - e^{0})) = 1$

Step 1

Simplify each term.

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Step 1.1

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

$x \frac{1}{e^{8}} - e^{- 8} - (x e^{0} - e^{0}) = 1$

Step 1.2

Combine $x$ and $\frac{1}{e^{8}}$ .

$\frac{x}{e^{8}} - e^{- 8} - (x e^{0} - e^{0}) = 1$

Step 1.3

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

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - (x e^{0} - e^{0}) = 1$

Step 1.4

Simplify each term.

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Step 1.4.1

Anything raised to $0$ is $1$ .

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - (x \cdot 1 - e^{0}) = 1$

Step 1.4.2

Multiply $x$ by $1$ .

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - (x - e^{0}) = 1$

Step 1.4.3

Anything raised to $0$ is $1$ .

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - (x - 1 \cdot 1) = 1$

Step 1.4.4

Multiply $- 1$ by $1$ .

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - (x - 1) = 1$

Step 1.5

Apply the distributive property.

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - x - - 1 = 1$

Step 1.6

Multiply $- 1$ by $- 1$ .

$\frac{x}{e^{8}} - \frac{1}{e^{8}} - x + 1 = 1$

Step 2

Multiply each term in $\frac{x}{e^{8}} - \frac{1}{e^{8}} - x + 1 = 1$ by $e^{8}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x}{e^{8}} - \frac{1}{e^{8}} - x + 1 = 1$ by $e^{8}$ .

$\frac{x}{e^{8}} e^{8} - \frac{1}{e^{8}} e^{8} - x e^{8} + 1 e^{8} = 1 e^{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $e^{8}$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x}{e^{8}} e^{8} - \frac{1}{e^{8}} e^{8} - x e^{8} + 1 e^{8} = 1 e^{8}$

Step 2.2.1.1.2

Rewrite the expression.

$x - \frac{1}{e^{8}} e^{8} - x e^{8} + 1 e^{8} = 1 e^{8}$

Step 2.2.1.2

Cancel the common factor of $e^{8}$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{1}{e^{8}}$ into the numerator.

$x + \frac{- 1}{e^{8}} e^{8} - x e^{8} + 1 e^{8} = 1 e^{8}$

Step 2.2.1.2.2

Cancel the common factor.

$x + \frac{- 1}{e^{8}} e^{8} - x e^{8} + 1 e^{8} = 1 e^{8}$

Step 2.2.1.2.3

Rewrite the expression.

$x - 1 - x e^{8} + 1 e^{8} = 1 e^{8}$

Step 2.2.1.3

Multiply $e^{8}$ by $1$ .

$x - 1 - x e^{8} + e^{8} = 1 e^{8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $e^{8}$ by $1$ .

$x - 1 - x e^{8} + e^{8} = e^{8}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Add $1$ to both sides of the equation.

$x - x e^{8} + e^{8} = e^{8} + 1$

Step 3.2

Subtract $e^{8}$ from both sides of the equation.

$x - x e^{8} = e^{8} + 1 - e^{8}$

Step 3.3

Subtract $e^{8}$ from $e^{8}$ .

$x - x e^{8} = 0 + 1$

Step 3.4

Add $0$ and $1$ .

$x - x e^{8} = 1$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $x$ out of $x - x e^{8}$ .

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Step 4.1.1

Raise $x$ to the power of $1$ .

$x - x e^{8} = 1$

Step 4.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x e^{8} = 1$

Step 4.1.3

Factor $x$ out of $- x e^{8}$ .

$x \cdot 1 + x (- 1 e^{8}) = 1$

Step 4.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 e^{8})$ .

$x (1 - 1 e^{8}) = 1$

Step 4.2

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

$x (1^{2} - 1 e^{8}) = 1$

Step 4.3

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$x (1^{2} - {(e^{4})}^{2}) = 1$

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{4}$ .

$x ((1 + e^{4}) (1 - e^{4})) = 1$

Step 4.5

Factor.

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Step 4.5.1

Simplify.

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Step 4.5.1.1

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

$x ((1 + e^{4}) (1^{2} - e^{4})) = 1$

Step 4.5.1.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

$x ((1 + e^{4}) (1^{2} - {(e^{2})}^{2})) = 1$

Step 4.5.1.3

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

$x ((1 + e^{4}) ((1 + e^{2}) (1 - e^{2}))) = 1$

Step 4.5.1.4

Factor.

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Step 4.5.1.4.1

Simplify.

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Step 4.5.1.4.1.1

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

$x ((1 + e^{4}) ((1 + e^{2}) (1^{2} - e^{2}))) = 1$

Step 4.5.1.4.1.2

Factor.

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Step 4.5.1.4.1.2.1

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

$x ((1 + e^{4}) ((1 + e^{2}) ((1 + e) (1 - e)))) = 1$

Step 4.5.1.4.1.2.2

Remove unnecessary parentheses.

$x ((1 + e^{4}) ((1 + e^{2}) (1 + e) (1 - e))) = 1$

Step 4.5.1.4.2

Remove unnecessary parentheses.

$x ((1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)) = 1$

Step 4.5.2

Remove unnecessary parentheses.

$x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e) = 1$

Step 5

Divide each term in $x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e) = 1$ by $1 - e^{8}$ and simplify.

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Step 5.1

Divide each term in $x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e) = 1$ by $1 - e^{8}$ .

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{1 - e^{8}} = \frac{1}{1 - e^{8}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify the denominator.

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Step 5.2.1.1

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

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{1^{2} - e^{8}} = \frac{1}{1 - e^{8}}$

Step 5.2.1.2

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{1^{2} - {(e^{4})}^{2}} = \frac{1}{1 - e^{8}}$

Step 5.2.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{4}$ .

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) (1 - e^{4})} = \frac{1}{1 - e^{8}}$

Step 5.2.1.4

Simplify.

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Step 5.2.1.4.1

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

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) (1^{2} - e^{4})} = \frac{1}{1 - e^{8}}$

Step 5.2.1.4.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) (1^{2} - {(e^{2})}^{2})} = \frac{1}{1 - e^{8}}$

Step 5.2.1.4.3

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

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) ((1 + e^{2}) (1 - e^{2}))} = \frac{1}{1 - e^{8}}$

Step 5.2.1.4.4

Simplify.

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Step 5.2.1.4.4.1

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

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) ((1 + e^{2}) (1^{2} - e^{2}))} = \frac{1}{1 - e^{8}}$

Step 5.2.1.4.4.2

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

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) ((1 + e^{2}) (1 + e) (1 - e))} = \frac{1}{1 - e^{8}}$

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)} = \frac{1}{1 - e^{8}}$

Step 5.2.2

Reduce the expression by cancelling the common factors.

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Step 5.2.2.1

Cancel the common factor of $1 + e^{4}$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{x (1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)} = \frac{1}{1 - e^{8}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{((x (1 + e^{2})) (1 + e)) (1 - e)}{((1 + e^{2}) (1 + e)) (1 - e)} = \frac{1}{1 - e^{8}}$

Step 5.2.2.2

Cancel the common factor of $1 + e^{2}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{x (1 + e^{2}) (1 + e) (1 - e)}{(1 + e^{2}) (1 + e) (1 - e)} = \frac{1}{1 - e^{8}}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{((x) (1 + e)) (1 - e)}{(1 + e) (1 - e)} = \frac{1}{1 - e^{8}}$

Step 5.2.2.3

Cancel the common factor of $1 + e$ .

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{x (1 + e) (1 - e)}{(1 + e) (1 - e)} = \frac{1}{1 - e^{8}}$

Step 5.2.2.3.2

Rewrite the expression.

$\frac{(x) (1 - e)}{1 - e} = \frac{1}{1 - e^{8}}$

Step 5.2.2.4

Cancel the common factor of $1 - e$ .

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Step 5.2.2.4.1

Cancel the common factor.

$\frac{x (1 - e)}{1 - e} = \frac{1}{1 - e^{8}}$

Step 5.2.2.4.2

Divide $x$ by $1$ .

$x = \frac{1}{1 - e^{8}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify the denominator.

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Step 5.3.1.1

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

$x = \frac{1}{1^{2} - e^{8}}$

Step 5.3.1.2

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$x = \frac{1}{1^{2} - {(e^{4})}^{2}}$

Step 5.3.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{4}$ .

$x = \frac{1}{(1 + e^{4}) (1 - e^{4})}$

Step 5.3.1.4

Simplify.

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Step 5.3.1.4.1

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

$x = \frac{1}{(1 + e^{4}) (1^{2} - e^{4})}$

Step 5.3.1.4.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

$x = \frac{1}{(1 + e^{4}) (1^{2} - {(e^{2})}^{2})}$

Step 5.3.1.4.3

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

$x = \frac{1}{(1 + e^{4}) ((1 + e^{2}) (1 - e^{2}))}$

Step 5.3.1.4.4

Simplify.

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Step 5.3.1.4.4.1

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

$x = \frac{1}{(1 + e^{4}) ((1 + e^{2}) (1^{2} - e^{2}))}$

Step 5.3.1.4.4.2

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

$x = \frac{1}{(1 + e^{4}) ((1 + e^{2}) (1 + e) (1 - e))}$

$x = \frac{1}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$

Decimal Form:

$x = - 0.00033557 \dots$