Basic Math Examples

$8400 = 5000 {(1 + \frac{r}{4})}^{32}$

Step 1

Rewrite the equation as $5000 {(1 + \frac{r}{4})}^{32} = 8400$ .

$5000 {(1 + \frac{r}{4})}^{32} = 8400$

Step 2

Divide each term in $5000 {(1 + \frac{r}{4})}^{32} = 8400$ by $5000$ and simplify.

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Step 2.1

Divide each term in $5000 {(1 + \frac{r}{4})}^{32} = 8400$ by $5000$ .

$\frac{5000 {(1 + \frac{r}{4})}^{32}}{5000} = \frac{8400}{5000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5000 {(1 + \frac{r}{4})}^{32}}{5000} = \frac{8400}{5000}$

Step 2.2.1.2

Divide ${(1 + \frac{r}{4})}^{32}$ by $1$ .

${(1 + \frac{r}{4})}^{32} = \frac{8400}{5000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $8400$ and $5000$ .

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Step 2.3.1.1

Factor $200$ out of $8400$ .

${(1 + \frac{r}{4})}^{32} = \frac{200 (42)}{5000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $200$ out of $5000$ .

${(1 + \frac{r}{4})}^{32} = \frac{200 \cdot 42}{200 \cdot 25}$

Step 2.3.1.2.2

Cancel the common factor.

${(1 + \frac{r}{4})}^{32} = \frac{200 \cdot 42}{200 \cdot 25}$

Step 2.3.1.2.3

Rewrite the expression.

${(1 + \frac{r}{4})}^{32} = \frac{42}{25}$

Step 3

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

$1 + \frac{r}{4} = \pm \sqrt[32]{\frac{42}{25}}$

Step 4

Simplify $\pm \sqrt[32]{\frac{42}{25}}$ .

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Step 4.1

Rewrite $\sqrt[32]{\frac{42}{25}}$ as $\frac{\sqrt[32]{42}}{\sqrt[32]{25}}$ .

$1 + \frac{r}{4} = \pm \frac{\sqrt[32]{42}}{\sqrt[32]{25}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Rewrite $25$ as $5^{2}$ .

$1 + \frac{r}{4} = \pm \frac{\sqrt[32]{42}}{\sqrt[32]{5^{2}}}$

Step 4.2.2

Rewrite $\sqrt[32]{5^{2}}$ as $\sqrt[16]{\sqrt{5^{2}}}$ .

$1 + \frac{r}{4} = \pm \frac{\sqrt[32]{42}}{\sqrt[16]{\sqrt{5^{2}}}}$

Step 4.2.3

Pull terms out from under the radical, assuming positive real numbers.

$1 + \frac{r}{4} = \pm \frac{\sqrt[32]{42}}{\sqrt[16]{5}}$

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.

$1 + \frac{r}{4} = \frac{\sqrt[32]{42}}{\sqrt[16]{5}}$

Step 5.2

Subtract $1$ from both sides of the equation.

$\frac{r}{4} = \frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1$

Step 5.3

Multiply both sides of the equation by $4$ .

$4 \frac{r}{4} = 4 (\frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$

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.

$4 \frac{r}{4} = 4 (\frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$

Step 5.4.1.1.2

Rewrite the expression.

$r = 4 (\frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $4 (\frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$ .

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Step 5.4.2.1.1

Apply the distributive property.

$r = 4 \frac{\sqrt[32]{42}}{\sqrt[16]{5}} + 4 \cdot - 1$

Step 5.4.2.1.2

Combine $4$ and $\frac{\sqrt[32]{42}}{\sqrt[16]{5}}$ .

$r = \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} + 4 \cdot - 1$

Step 5.4.2.1.3

Multiply $4$ by $- 1$ .

$r = \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4$

Step 5.5

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

$1 + \frac{r}{4} = - \frac{\sqrt[32]{42}}{\sqrt[16]{5}}$

Step 5.6

Subtract $1$ from both sides of the equation.

$\frac{r}{4} = - \frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1$

Step 5.7

Multiply both sides of the equation by $4$ .

$4 \frac{r}{4} = 4 (- \frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$

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.

$4 \frac{r}{4} = 4 (- \frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$

Step 5.8.1.1.2

Rewrite the expression.

$r = 4 (- \frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$

Step 5.8.2

Simplify the right side.

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Step 5.8.2.1

Simplify $4 (- \frac{\sqrt[32]{42}}{\sqrt[16]{5}} - 1)$ .

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Step 5.8.2.1.1

Apply the distributive property.

$r = 4 (- \frac{\sqrt[32]{42}}{\sqrt[16]{5}}) + 4 \cdot - 1$

Step 5.8.2.1.2

Multiply $4 (- \frac{\sqrt[32]{42}}{\sqrt[16]{5}})$ .

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Step 5.8.2.1.2.1

Multiply $- 1$ by $4$ .

$r = - 4 \frac{\sqrt[32]{42}}{\sqrt[16]{5}} + 4 \cdot - 1$

Step 5.8.2.1.2.2

Combine $- 4$ and $\frac{\sqrt[32]{42}}{\sqrt[16]{5}}$ .

$r = \frac{- 4 \sqrt[32]{42}}{\sqrt[16]{5}} + 4 \cdot - 1$

Step 5.8.2.1.3

Multiply $4$ by $- 1$ .

$r = \frac{- 4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4$

Step 5.8.2.1.4

Move the negative in front of the fraction.

$r = - \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4$

Step 5.9

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

$r = \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4, - \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4, - \frac{4 \sqrt[32]{42}}{\sqrt[16]{5}} - 4$

Decimal Form:

$r = 0.06537775 \dots, - 8.06537775 \dots$