Calculus Examples

$e - e^{- 1}$

Step 1

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

$e - \frac{1}{e}$

Step 2

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

$\frac{e e}{e} - \frac{1}{e}$

Step 3

Combine the numerators over the common denominator.

$\frac{e e - 1}{e}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $e e$ .

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Step 4.1.1

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

$\frac{e^{1} e - 1}{e}$

Step 4.1.2

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

$\frac{e^{1} e^{1} - 1}{e}$

Step 4.1.3

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

$\frac{e^{1 + 1} - 1}{e}$

Step 4.1.4

Add $1$ and $1$ .

$\frac{e^{2} - 1}{e}$

Step 4.2

Rewrite $e^{2} - 1$ in a factored form.

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Step 4.2.1

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

$\frac{e^{2} - 1^{2}}{e}$

Step 4.2.2

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.35040238 \dots$