Finite Math Examples

$0.0658 = {(1 + \frac{r}{4})}^{4} - 1$

Step 1

Rewrite the equation as ${(1 + \frac{r}{4})}^{4} - 1 = 0.0658$ .

${(1 + \frac{r}{4})}^{4} - 1 = 0.0658$

Step 2

Add $1$ to both sides of the equation.

${(1 + \frac{r}{4})}^{4} = 0.0658 + 1$

Step 3

Add $0.0658$ and $1$ .

${(1 + \frac{r}{4})}^{4} = 1.0658$

Step 4

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

$1 + \frac{r}{4} = \pm \sqrt[4]{1.0658}$

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.

$1 + \frac{r}{4} = \sqrt[4]{1.0658}$

Step 5.2

Subtract $1$ from both sides of the equation.

$\frac{r}{4} = \sqrt[4]{1.0658} - 1$

Step 5.3

Multiply both sides of the equation by $4$ .

$4 \frac{r}{4} = 4 (\sqrt[4]{1.0658} - 1)$

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 $4$ .

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Step 5.4.1.1.1

Cancel the common factor.

$4 \frac{r}{4} = 4 (\sqrt[4]{1.0658} - 1)$

Step 5.4.1.1.2

Rewrite the expression.

$r = 4 (\sqrt[4]{1.0658} - 1)$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $4 (\sqrt[4]{1.0658} - 1)$ .

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Step 5.4.2.1.1

Apply the distributive property.

$r = 4 \sqrt[4]{1.0658} + 4 \cdot - 1$

Step 5.4.2.1.2

Multiply $4$ by $- 1$ .

$r = 4 \sqrt[4]{1.0658} - 4$

Step 5.5

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

$1 + \frac{r}{4} = - \sqrt[4]{1.0658}$

Step 5.6

Subtract $1$ from both sides of the equation.

$\frac{r}{4} = - \sqrt[4]{1.0658} - 1$

Step 5.7

Multiply both sides of the equation by $4$ .

$4 \frac{r}{4} = 4 (- \sqrt[4]{1.0658} - 1)$

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 $4$ .

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Step 5.8.1.1.1

Cancel the common factor.

$4 \frac{r}{4} = 4 (- \sqrt[4]{1.0658} - 1)$

Step 5.8.1.1.2

Rewrite the expression.

$r = 4 (- \sqrt[4]{1.0658} - 1)$

Step 5.8.2

Simplify the right side.

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Step 5.8.2.1

Simplify $4 (- \sqrt[4]{1.0658} - 1)$ .

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Step 5.8.2.1.1

Apply the distributive property.

$r = 4 (- \sqrt[4]{1.0658}) + 4 \cdot - 1$

Step 5.8.2.1.2

Multiply.

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Step 5.8.2.1.2.1

Multiply $- 1$ by $4$ .

$r = - 4 \sqrt[4]{1.0658} + 4 \cdot - 1$

Step 5.8.2.1.2.2

Multiply $4$ by $- 1$ .

$r = - 4 \sqrt[4]{1.0658} - 4$

Step 5.9

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

$r = 4 \sqrt[4]{1.0658} - 4, - 4 \sqrt[4]{1.0658} - 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = 4 \sqrt[4]{1.0658} - 4, - 4 \sqrt[4]{1.0658} - 4$

Decimal Form:

$r = 0.06423601 \dots, - 8.06423601 \dots$