Finite Math Examples

$e^{0.15} - 1 = {(1 + \frac{r}{12})}^{12} - 1$

Step 1

Rewrite the equation as ${(1 + \frac{r}{12})}^{12} - 1 = e^{0.15} - 1$ .

${(1 + \frac{r}{12})}^{12} - 1 = e^{0.15} - 1$

Step 2

Add $1$ to both sides of the equation.

${(1 + \frac{r}{12})}^{12} = e^{0.15} - 1 + 1$

Step 3

Add $- 1$ and $1$ .

${(1 + \frac{r}{12})}^{12} = e^{0.15} + 0$

Step 4

Add $e^{0.15}$ and $0$ .

${(1 + \frac{r}{12})}^{12} = e^{0.15}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$1 + \frac{r}{12} = \pm \sqrt[12]{e^{0.15}}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$1 + \frac{r}{12} = \sqrt[12]{e^{0.15}}$

Step 6.2

Subtract $1$ from both sides of the equation.

$\frac{r}{12} = \sqrt[12]{e^{0.15}} - 1$

Step 6.3

Multiply both sides of the equation by $12$ .

$12 \frac{r}{12} = 12 (\sqrt[12]{e^{0.15}} - 1)$

Step 6.4

Simplify both sides of the equation.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Cancel the common factor of $12$ .

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Step 6.4.1.1.1

Cancel the common factor.

$12 \frac{r}{12} = 12 (\sqrt[12]{e^{0.15}} - 1)$

Step 6.4.1.1.2

Rewrite the expression.

$r = 12 (\sqrt[12]{e^{0.15}} - 1)$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $12 (\sqrt[12]{e^{0.15}} - 1)$ .

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Step 6.4.2.1.1

Apply the distributive property.

$r = 12 \sqrt[12]{e^{0.15}} + 12 \cdot - 1$

Step 6.4.2.1.2

Multiply $12$ by $- 1$ .

$r = 12 \sqrt[12]{e^{0.15}} - 12$

Step 6.5

Next, use the negative value of the $\pm$ to find the second solution.

$1 + \frac{r}{12} = - \sqrt[12]{e^{0.15}}$

Step 6.6

Subtract $1$ from both sides of the equation.

$\frac{r}{12} = - \sqrt[12]{e^{0.15}} - 1$

Step 6.7

Multiply both sides of the equation by $12$ .

$12 \frac{r}{12} = 12 (- \sqrt[12]{e^{0.15}} - 1)$

Step 6.8

Simplify both sides of the equation.

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Step 6.8.1

Simplify the left side.

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Step 6.8.1.1

Cancel the common factor of $12$ .

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Step 6.8.1.1.1

Cancel the common factor.

$12 \frac{r}{12} = 12 (- \sqrt[12]{e^{0.15}} - 1)$

Step 6.8.1.1.2

Rewrite the expression.

$r = 12 (- \sqrt[12]{e^{0.15}} - 1)$

Step 6.8.2

Simplify the right side.

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Step 6.8.2.1

Simplify $12 (- \sqrt[12]{e^{0.15}} - 1)$ .

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Step 6.8.2.1.1

Apply the distributive property.

$r = 12 (- \sqrt[12]{e^{0.15}}) + 12 \cdot - 1$

Step 6.8.2.1.2

Multiply.

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Step 6.8.2.1.2.1

Multiply $- 1$ by $12$ .

$r = - 12 \sqrt[12]{e^{0.15}} + 12 \cdot - 1$

Step 6.8.2.1.2.2

Multiply $12$ by $- 1$ .

$r = - 12 \sqrt[12]{e^{0.15}} - 12$

Step 6.9

The complete solution is the result of both the positive and negative portions of the solution.

$r = 12 \sqrt[12]{e^{0.15}} - 12, - 12 \sqrt[12]{e^{0.15}} - 12$

Step 7

The result can be shown in multiple forms.

Exact Form:

$r = 12 \sqrt[12]{e^{0.15}} - 12, - 12 \sqrt[12]{e^{0.15}} - 12$

Decimal Form:

$r = 0.15094141 \dots, - 24.15094141 \dots$