Calculus Examples

$\frac{8}{9} \cdot e^{9} - \frac{8}{9}$

Step 1

Simplify terms.

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Step 1.1

Combine

$\frac{8}{9}$ and

$e^{9}$ .

$\frac{8 e^{9}}{9} - \frac{8}{9}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{8 e^{9} - 8}{9}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

$8 e^{9}$ as

${(2 e^{3})}^{3}$ .

$\frac{{(2 e^{3})}^{3} - 8}{9}$

Step 2.2

Rewrite

$8$ as

$2^{3}$ .

$\frac{{(2 e^{3})}^{3} - 2^{3}}{9}$

Step 2.3

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

$a = 2 e^{3}$ and

$b = 2$ .

$\frac{(2 e^{3} - 2) ({(2 e^{3})}^{2} + 2 e^{3} \cdot 2 + 2^{2})}{9}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to

$2 e^{3}$ .

$\frac{(2 e^{3} - 2) (2^{2} {(e^{3})}^{2} + 2 e^{3} \cdot 2 + 2^{2})}{9}$

Step 2.4.2

Raise

$2$ to the power of

$2$ .

$\frac{(2 e^{3} - 2) (4 {(e^{3})}^{2} + 2 e^{3} \cdot 2 + 2^{2})}{9}$

Step 2.4.3

Multiply the exponents in

${(e^{3})}^{2}$ .

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Step 2.4.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{(2 e^{3} - 2) (4 e^{3 \cdot 2} + 2 e^{3} \cdot 2 + 2^{2})}{9}$

Step 2.4.3.2

Multiply

$3$ by

$2$ .

$\frac{(2 e^{3} - 2) (4 e^{6} + 2 e^{3} \cdot 2 + 2^{2})}{9}$

Step 2.4.4

Multiply

$2$ by

$2$ .

$\frac{(2 e^{3} - 2) (4 e^{6} + 4 e^{3} + 2^{2})}{9}$

Step 2.4.5

Raise

$2$ to the power of

$2$ .

$\frac{(2 e^{3} - 2) (4 e^{6} + 4 e^{3} + 4)}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{(2 e^{3} - 2) (4 e^{6} + 4 e^{3} + 4)}{9}$

Decimal Form:

$7201.85238006 \dots$