Algebra Examples

$1 - e^{- 3}$

Step 1

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

$1 - \frac{1}{e^{3}}$

Step 2

Simplify the expression.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$\frac{e^{3}}{e^{3}} - \frac{1}{e^{3}}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{e^{3} - 1}{e^{3}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $1$ as $1^{3}$ .

$\frac{e^{3} - 1^{3}}{e^{3}}$

Step 3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = e$ and $b = 1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Multiply $e$ by $1$ .

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

Step 3.3.2

One to any power is one.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.95021293 \dots$