Calculus Examples

$500 e \cdot (- \frac{1}{0.05} \cdot e^{- 0.05 \cdot 25} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1

Simplify each term.

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Step 1.1

Divide $1$ by $0.05$ .

$500 e \cdot (- 1 \cdot 20 \cdot e^{- 0.05 \cdot 25} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1.2

Multiply $- 1$ by $20$ .

$500 e \cdot (- 20 \cdot e^{- 0.05 \cdot 25} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1.3

Multiply $- 0.05$ by $25$ .

$500 e \cdot (- 20 \cdot e^{- 1.25} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1.4

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

$500 e \cdot (- 20 \cdot \frac{1}{e^{1.25}} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1.5

Combine $- 20$ and $\frac{1}{e^{1.25}}$ .

$500 e \cdot (\frac{- 20}{e^{1.25}} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1.6

Move the negative in front of the fraction.

$500 e \cdot (- \frac{20}{e^{1.25}} + \frac{1}{0.05} \cdot e^{- 0.05 \cdot 0})$

Step 1.7

Divide $1$ by $0.05$ .

$500 e \cdot (- \frac{20}{e^{1.25}} + 20 \cdot e^{- 0.05 \cdot 0})$

Step 1.8

Multiply $- 0.05$ by $0$ .

$500 e \cdot (- \frac{20}{e^{1.25}} + 20 \cdot e^{0})$

Step 1.9

Anything raised to $0$ is $1$ .

$500 e \cdot (- \frac{20}{e^{1.25}} + 20 \cdot 1)$

Step 1.10

Multiply $20$ by $1$ .

$500 e \cdot (- \frac{20}{e^{1.25}} + 20)$

Step 2

To write $20$ as a fraction with a common denominator, multiply by $\frac{e^{1.25}}{e^{1.25}}$ .

$500 e \cdot (- \frac{20}{e^{1.25}} + \frac{20 e^{1.25}}{e^{1.25}})$

Step 3

Combine the numerators over the common denominator.

$500 e \cdot \frac{- 20 + 20 e^{1.25}}{e^{1.25}}$

Step 4

Multiply $500 e \frac{- 20 + 20 e^{1.25}}{e^{1.25}}$ .

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Step 4.1

Combine $\frac{- 20 + 20 e^{1.25}}{e^{1.25}}$ and $500$ .

$\frac{(- 20 + 20 e^{1.25}) \cdot 500}{e^{1.25}} e$

Step 4.2

Combine $\frac{(- 20 + 20 e^{1.25}) \cdot 500}{e^{1.25}}$ and $e$ .

$\frac{(- 20 + 20 e^{1.25}) \cdot 500 e}{e^{1.25}}$

Step 5

Move $500$ to the left of $- 20 + 20 e^{1.25}$ .

$\frac{500 \cdot (- 20 + 20 e^{1.25}) e}{e^{1.25}}$

Step 6

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{500 (- 1 (20) + 20 e^{1.25}) e}{e^{1.25}}$

Step 7

Factor $- 1$ out of $20 e^{1.25}$ .

$\frac{500 (- 1 (20) - (- 20 e^{1.25})) e}{e^{1.25}}$

Step 8

Factor $- 1$ out of $- 1 (20) - (- 20 e^{1.25})$ .

$\frac{500 (- 1 (20 - 20 e^{1.25})) e}{e^{1.25}}$

Step 9

Simplify the expression.

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Step 9.1

Move the negative in front of the fraction.

$- \frac{(500 (20 - 20 e^{1.25})) e}{e^{1.25}}$

Step 9.2

Reorder factors in $- \frac{(500 (20 - 20 e^{1.25})) e}{e^{1.25}}$ .

$- \frac{(500) e (20 - 20 e^{1.25})}{e^{1.25}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{(500) e (20 - 20 e^{1.25})}{e^{1.25}}$

Decimal Form:

$19394.81045387 \dots$