Calculus Examples

$P (14) = 5000 e^{\frac{\ln (\frac{29}{25})}{2} (14)}$

Step 1

Simplify.

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Step 1.1

Cancel the common factor of $2$ .

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Step 1.1.1

Factor $2$ out of $14$ .

$P (14) = 5000 e^{\frac{\ln (\frac{29}{25})}{2} \cdot (2 (7))}$

Step 1.1.2

Cancel the common factor.

$P (14) = 5000 e^{\frac{\ln (\frac{29}{25})}{2} \cdot (2 \cdot 7)}$

Step 1.1.3

Rewrite the expression.

$P (14) = 5000 e^{\ln (\frac{29}{25}) \cdot 7}$

Step 1.2

Move $7$ to the left of $\ln (\frac{29}{25})$ .

$P (14) = 5000 e^{7 \cdot \ln (\frac{29}{25})}$

Step 1.3

Simplify $7 \ln (\frac{29}{25})$ by moving $7$ inside the logarithm.

$P (14) = 5000 e^{\ln ({(\frac{29}{25})}^{7})}$

Step 1.4

Exponentiation and log are inverse functions.

$P (14) = 5000 {(\frac{29}{25})}^{7}$

Step 1.5

Apply the product rule to $\frac{29}{25}$ .

$P (14) = 5000 \frac{29^{7}}{25^{7}}$

Step 1.6

Raise $29$ to the power of $7$ .

$P (14) = 5000 \frac{17249876309}{25^{7}}$

Step 1.7

Raise $25$ to the power of $7$ .

$P (14) = 5000 (\frac{17249876309}{6103515625})$

Step 1.8

Cancel the common factor of $625$ .

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Step 1.8.1

Factor $625$ out of $5000$ .

$P (14) = 625 (8) \frac{17249876309}{6103515625}$

Step 1.8.2

Factor $625$ out of $6103515625$ .

$P (14) = 625 \cdot 8 \frac{17249876309}{625 \cdot 9765625}$

Step 1.8.3

Cancel the common factor.

$P (14) = 625 \cdot 8 \frac{17249876309}{625 \cdot 9765625}$

Step 1.8.4

Rewrite the expression.

$P (14) = 8 (\frac{17249876309}{9765625})$

Step 1.9

Combine $8$ and $\frac{17249876309}{9765625}$ .

$P (14) = \frac{8 \cdot 17249876309}{9765625}$

Step 1.10

Multiply $8$ by $17249876309$ .

$P (14) = \frac{137999010472}{9765625}$

Step 2

Divide each term in $P (14) = \frac{137999010472}{9765625}$ by $14$ and simplify.

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Step 2.1

Divide each term in $P (14) = \frac{137999010472}{9765625}$ by $14$ .

$\frac{P (14)}{14} = \frac{\frac{137999010472}{9765625}}{14}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $14$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{P \cdot 14}{14} = \frac{\frac{137999010472}{9765625}}{14}$

Step 2.2.1.2

Divide $P$ by $1$ .

$P = \frac{\frac{137999010472}{9765625}}{14}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

$P = \frac{137999010472}{9765625} \cdot \frac{1}{14}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Factor $2$ out of $137999010472$ .

$P = \frac{2 (68999505236)}{9765625} \cdot \frac{1}{14}$

Step 2.3.2.2

Factor $2$ out of $14$ .

$P = \frac{2 \cdot 68999505236}{9765625} \cdot \frac{1}{2 \cdot 7}$

Step 2.3.2.3

Cancel the common factor.

$P = \frac{2 \cdot 68999505236}{9765625} \cdot \frac{1}{2 \cdot 7}$

Step 2.3.2.4

Rewrite the expression.

$P = \frac{68999505236}{9765625} \cdot \frac{1}{7}$

Step 2.3.3

Multiply $\frac{68999505236}{9765625}$ by $\frac{1}{7}$ .

$P = \frac{68999505236}{9765625 \cdot 7}$

Step 2.3.4

Multiply $9765625$ by $7$ .

$P = \frac{68999505236}{68359375}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$P = \frac{68999505236}{68359375}$

Decimal Form:

$P = 1009.36419088 \dots$

Mixed Number Form:

$P = 1009 \frac{24895861}{68359375}$