Finite Math Examples

$0.0429 = {(1 + \frac{r}{12})}^{12} - 1$

Step 1

Rewrite the equation as ${(1 + \frac{r}{12})}^{12} - 1 = 0.0429$ .

${(1 + \frac{r}{12})}^{12} - 1 = 0.0429$

Step 2

Add $1$ to both sides of the equation.

${(1 + \frac{r}{12})}^{12} = 0.0429 + 1$

Step 3

Add $0.0429$ and $1$ .

${(1 + \frac{r}{12})}^{12} = 1.0429$

Step 4

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

$1 + \frac{r}{12} = \pm \sqrt[12]{1.0429}$

Step 5

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

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Step 5.1

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

$1 + \frac{r}{12} = \sqrt[12]{1.0429}$

Step 5.2

Subtract $1$ from both sides of the equation.

$\frac{r}{12} = \sqrt[12]{1.0429} - 1$

Step 5.3

Multiply both sides of the equation by $12$ .

$12 \frac{r}{12} = 12 (\sqrt[12]{1.0429} - 1)$

Step 5.4

Simplify both sides of the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Cancel the common factor of $12$ .

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Step 5.4.1.1.1

Cancel the common factor.

$12 \frac{r}{12} = 12 (\sqrt[12]{1.0429} - 1)$

Step 5.4.1.1.2

Rewrite the expression.

$r = 12 (\sqrt[12]{1.0429} - 1)$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $12 (\sqrt[12]{1.0429} - 1)$ .

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Step 5.4.2.1.1

Apply the distributive property.

$r = 12 \sqrt[12]{1.0429} + 12 \cdot - 1$

Step 5.4.2.1.2

Multiply $12$ by $- 1$ .

$r = 12 \sqrt[12]{1.0429} - 12$

Step 5.5

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

$1 + \frac{r}{12} = - \sqrt[12]{1.0429}$

Step 5.6

Subtract $1$ from both sides of the equation.

$\frac{r}{12} = - \sqrt[12]{1.0429} - 1$

Step 5.7

Multiply both sides of the equation by $12$ .

$12 \frac{r}{12} = 12 (- \sqrt[12]{1.0429} - 1)$

Step 5.8

Simplify both sides of the equation.

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Step 5.8.1

Simplify the left side.

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Step 5.8.1.1

Cancel the common factor of $12$ .

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Step 5.8.1.1.1

Cancel the common factor.

$12 \frac{r}{12} = 12 (- \sqrt[12]{1.0429} - 1)$

Step 5.8.1.1.2

Rewrite the expression.

$r = 12 (- \sqrt[12]{1.0429} - 1)$

Step 5.8.2

Simplify the right side.

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Step 5.8.2.1

Simplify $12 (- \sqrt[12]{1.0429} - 1)$ .

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Step 5.8.2.1.1

Apply the distributive property.

$r = 12 (- \sqrt[12]{1.0429}) + 12 \cdot - 1$

Step 5.8.2.1.2

Multiply.

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Step 5.8.2.1.2.1

Multiply $- 1$ by $12$ .

$r = - 12 \sqrt[12]{1.0429} + 12 \cdot - 1$

Step 5.8.2.1.2.2

Multiply $12$ by $- 1$ .

$r = - 12 \sqrt[12]{1.0429} - 12$

Step 5.9

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

$r = 12 \sqrt[12]{1.0429} - 12, - 12 \sqrt[12]{1.0429} - 12$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = 12 \sqrt[12]{1.0429} - 12, - 12 \sqrt[12]{1.0429} - 12$

Decimal Form:

$r = 0.04207889 \dots, - 24.04207889 \dots$