Calculus Examples

$7000 = 5000 {(1 + \frac{r}{12})}^{120}$

Step 1

Rewrite the equation as $5000 {(1 + \frac{r}{12})}^{120} = 7000$ .

$5000 {(1 + \frac{r}{12})}^{120} = 7000$

Step 2

Divide each term in $5000 {(1 + \frac{r}{12})}^{120} = 7000$ by $5000$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $5000 {(1 + \frac{r}{12})}^{120} = 7000$ by $5000$ .

$\frac{5000 {(1 + \frac{r}{12})}^{120}}{5000} = \frac{7000}{5000}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $5000$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{5000 {(1 + \frac{r}{12})}^{120}}{5000} = \frac{7000}{5000}$

Step 2.2.1.2

Divide ${(1 + \frac{r}{12})}^{120}$ by $1$ .

${(1 + \frac{r}{12})}^{120} = \frac{7000}{5000}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $7000$ and $5000$ .

Tap for more steps...

Step 2.3.1.1

Factor $1000$ out of $7000$ .

${(1 + \frac{r}{12})}^{120} = \frac{1000 (7)}{5000}$

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $1000$ out of $5000$ .

${(1 + \frac{r}{12})}^{120} = \frac{1000 \cdot 7}{1000 \cdot 5}$

Step 2.3.1.2.2

Cancel the common factor.

${(1 + \frac{r}{12})}^{120} = \frac{1000 \cdot 7}{1000 \cdot 5}$

Step 2.3.1.2.3

Rewrite the expression.

${(1 + \frac{r}{12})}^{120} = \frac{7}{5}$

Step 3

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

$1 + \frac{r}{12} = \pm \sqrt[120]{\frac{7}{5}}$

Step 4

Rewrite $\sqrt[120]{\frac{7}{5}}$ as $\frac{\sqrt[120]{7}}{\sqrt[120]{5}}$ .

$1 + \frac{r}{12} = \pm \frac{\sqrt[120]{7}}{\sqrt[120]{5}}$

Step 5

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

Tap for more steps...

Step 5.1

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

$1 + \frac{r}{12} = \frac{\sqrt[120]{7}}{\sqrt[120]{5}}$

Step 5.2

Subtract $1$ from both sides of the equation.

$\frac{r}{12} = \frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1$

Step 5.3

Multiply both sides of the equation by $12$ .

$12 \frac{r}{12} = 12 (\frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$

Step 5.4

Simplify both sides of the equation.

Tap for more steps...

Step 5.4.1

Simplify the left side.

Tap for more steps...

Step 5.4.1.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 5.4.1.1.1

Cancel the common factor.

$12 \frac{r}{12} = 12 (\frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$

Step 5.4.1.1.2

Rewrite the expression.

$r = 12 (\frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$

Step 5.4.2

Simplify the right side.

Tap for more steps...

Step 5.4.2.1

Simplify $12 (\frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$ .

Tap for more steps...

Step 5.4.2.1.1

Apply the distributive property.

$r = 12 \frac{\sqrt[120]{7}}{\sqrt[120]{5}} + 12 \cdot - 1$

Step 5.4.2.1.2

Combine $12$ and $\frac{\sqrt[120]{7}}{\sqrt[120]{5}}$ .

$r = \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} + 12 \cdot - 1$

Step 5.4.2.1.3

Multiply $12$ by $- 1$ .

$r = \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12$

Step 5.5

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

$1 + \frac{r}{12} = - \frac{\sqrt[120]{7}}{\sqrt[120]{5}}$

Step 5.6

Subtract $1$ from both sides of the equation.

$\frac{r}{12} = - \frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1$

Step 5.7

Multiply both sides of the equation by $12$ .

$12 \frac{r}{12} = 12 (- \frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$

Step 5.8

Simplify both sides of the equation.

Tap for more steps...

Step 5.8.1

Simplify the left side.

Tap for more steps...

Step 5.8.1.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 5.8.1.1.1

Cancel the common factor.

$12 \frac{r}{12} = 12 (- \frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$

Step 5.8.1.1.2

Rewrite the expression.

$r = 12 (- \frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$

Step 5.8.2

Simplify the right side.

Tap for more steps...

Step 5.8.2.1

Simplify $12 (- \frac{\sqrt[120]{7}}{\sqrt[120]{5}} - 1)$ .

Tap for more steps...

Step 5.8.2.1.1

Apply the distributive property.

$r = 12 (- \frac{\sqrt[120]{7}}{\sqrt[120]{5}}) + 12 \cdot - 1$

Step 5.8.2.1.2

Multiply $12 (- \frac{\sqrt[120]{7}}{\sqrt[120]{5}})$ .

Tap for more steps...

Step 5.8.2.1.2.1

Multiply $- 1$ by $12$ .

$r = - 12 \frac{\sqrt[120]{7}}{\sqrt[120]{5}} + 12 \cdot - 1$

Step 5.8.2.1.2.2

Combine $- 12$ and $\frac{\sqrt[120]{7}}{\sqrt[120]{5}}$ .

$r = \frac{- 12 \sqrt[120]{7}}{\sqrt[120]{5}} + 12 \cdot - 1$

Step 5.8.2.1.3

Multiply $12$ by $- 1$ .

$r = \frac{- 12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12$

Step 5.8.2.1.4

Move the negative in front of the fraction.

$r = - \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12$

Step 5.9

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

$r = \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12, - \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12, - \frac{12 \sqrt[120]{7}}{\sqrt[120]{5}} - 12$

Decimal Form:

$r = 0.03369444 \dots, - 24.03369444 \dots$