Basic Math Examples

$200 = 100 {(1 + \frac{r}{2})}^{8}$

Step 1

Rewrite the equation as $100 {(1 + \frac{r}{2})}^{8} = 200$ .

$100 {(1 + \frac{r}{2})}^{8} = 200$

Step 2

Divide each term in $100 {(1 + \frac{r}{2})}^{8} = 200$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 {(1 + \frac{r}{2})}^{8} = 200$ by $100$ .

$\frac{100 {(1 + \frac{r}{2})}^{8}}{100} = \frac{200}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 {(1 + \frac{r}{2})}^{8}}{100} = \frac{200}{100}$

Step 2.2.1.2

Divide ${(1 + \frac{r}{2})}^{8}$ by $1$ .

${(1 + \frac{r}{2})}^{8} = \frac{200}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $200$ by $100$ .

${(1 + \frac{r}{2})}^{8} = 2$

Step 3

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

$1 + \frac{r}{2} = \pm \sqrt[8]{2}$

Step 4

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

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Step 4.1

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

$1 + \frac{r}{2} = \sqrt[8]{2}$

Step 4.2

Subtract $1$ from both sides of the equation.

$\frac{r}{2} = \sqrt[8]{2} - 1$

Step 4.3

Multiply both sides of the equation by $2$ .

$2 \frac{r}{2} = 2 (\sqrt[8]{2} - 1)$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Cancel the common factor of $2$ .

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Step 4.4.1.1.1

Cancel the common factor.

$2 \frac{r}{2} = 2 (\sqrt[8]{2} - 1)$

Step 4.4.1.1.2

Rewrite the expression.

$r = 2 (\sqrt[8]{2} - 1)$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $2 (\sqrt[8]{2} - 1)$ .

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Step 4.4.2.1.1

Apply the distributive property.

$r = 2 \sqrt[8]{2} + 2 \cdot - 1$

Step 4.4.2.1.2

Multiply $2$ by $- 1$ .

$r = 2 \sqrt[8]{2} - 2$

Step 4.5

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

$1 + \frac{r}{2} = - \sqrt[8]{2}$

Step 4.6

Subtract $1$ from both sides of the equation.

$\frac{r}{2} = - \sqrt[8]{2} - 1$

Step 4.7

Multiply both sides of the equation by $2$ .

$2 \frac{r}{2} = 2 (- \sqrt[8]{2} - 1)$

Step 4.8

Simplify both sides of the equation.

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Step 4.8.1

Simplify the left side.

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Step 4.8.1.1

Cancel the common factor of $2$ .

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Step 4.8.1.1.1

Cancel the common factor.

$2 \frac{r}{2} = 2 (- \sqrt[8]{2} - 1)$

Step 4.8.1.1.2

Rewrite the expression.

$r = 2 (- \sqrt[8]{2} - 1)$

Step 4.8.2

Simplify the right side.

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Step 4.8.2.1

Simplify $2 (- \sqrt[8]{2} - 1)$ .

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Step 4.8.2.1.1

Apply the distributive property.

$r = 2 (- \sqrt[8]{2}) + 2 \cdot - 1$

Step 4.8.2.1.2

Multiply.

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Step 4.8.2.1.2.1

Multiply $- 1$ by $2$ .

$r = - 2 \sqrt[8]{2} + 2 \cdot - 1$

Step 4.8.2.1.2.2

Multiply $2$ by $- 1$ .

$r = - 2 \sqrt[8]{2} - 2$

Step 4.9

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

$r = 2 \sqrt[8]{2} - 2, - 2 \sqrt[8]{2} - 2$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = 2 \sqrt[8]{2} - 2, - 2 \sqrt[8]{2} - 2$

Decimal Form:

$r = 0.18101546 \dots, - 4.18101546 \dots$