Calculus Examples

${(- 6)}^{6} e^{- 6} - 8$

Step 1

Simplify each term.

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Step 1.1

Raise $- 6$ to the power of $6$ .

$46656 e^{- 6} - 8$

Step 1.2

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

$46656 \frac{1}{e^{6}} - 8$

Step 1.3

Combine $46656$ and $\frac{1}{e^{6}}$ .

$\frac{46656}{e^{6}} - 8$

Step 2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{e^{6}}{e^{6}}$ .

$\frac{46656}{e^{6}} - 8 \cdot \frac{e^{6}}{e^{6}}$

Step 3

Combine fractions.

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Step 3.1

Combine $- 8$ and $\frac{e^{6}}{e^{6}}$ .

$\frac{46656}{e^{6}} + \frac{- 8 e^{6}}{e^{6}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{46656 - 8 e^{6}}{e^{6}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $46656$ as $36^{3}$ .

$\frac{36^{3} - 8 e^{6}}{e^{6}}$

Step 4.2

Rewrite $8 e^{6}$ as ${(2 e^{2})}^{3}$ .

$\frac{36^{3} - {(2 e^{2})}^{3}}{e^{6}}$

Step 4.3

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 $a = 36$ and $b = 2 e^{2}$ .

$\frac{(36 - (2 e^{2})) (36^{2} + 36 (2 e^{2}) + {(2 e^{2})}^{2})}{e^{6}}$

Step 4.4

Simplify.

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Step 4.4.1

Multiply $2$ by $- 1$ .

$\frac{(36 - 2 e^{2}) (36^{2} + 36 (2 e^{2}) + {(2 e^{2})}^{2})}{e^{6}}$

Step 4.4.2

Raise $36$ to the power of $2$ .

$\frac{(36 - 2 e^{2}) (1296 + 36 (2 e^{2}) + {(2 e^{2})}^{2})}{e^{6}}$

Step 4.4.3

Multiply $2$ by $36$ .

$\frac{(36 - 2 e^{2}) (1296 + 72 e^{2} + {(2 e^{2})}^{2})}{e^{6}}$

Step 4.4.4

Apply the product rule to $2 e^{2}$ .

$\frac{(36 - 2 e^{2}) (1296 + 72 e^{2} + 2^{2} {(e^{2})}^{2})}{e^{6}}$

Step 4.4.5

Raise $2$ to the power of $2$ .

$\frac{(36 - 2 e^{2}) (1296 + 72 e^{2} + 4 {(e^{2})}^{2})}{e^{6}}$

Step 4.4.6

Multiply the exponents in ${(e^{2})}^{2}$ .

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Step 4.4.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(36 - 2 e^{2}) (1296 + 72 e^{2} + 4 e^{2 \cdot 2})}{e^{6}}$

Step 4.4.6.2

Multiply $2$ by $2$ .

$\frac{(36 - 2 e^{2}) (1296 + 72 e^{2} + 4 e^{4})}{e^{6}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{(36 - 2 e^{2}) (1296 + 72 e^{2} + 4 e^{4})}{e^{6}}$

Decimal Form:

$107.64866155 \dots$