Finite Math Examples

$4400 = 1100 {(1 + \frac{x}{6})}^{6 \cdot 3}$

Step 1

Rewrite the equation as $1100 {(1 + \frac{x}{6})}^{6 \cdot 3} = 4400$ .

$1100 {(1 + \frac{x}{6})}^{6 \cdot 3} = 4400$

Step 2

Multiply $6$ by $3$ .

$1100 {(1 + \frac{x}{6})}^{18} = 4400$

Step 3

Divide each term in $1100 {(1 + \frac{x}{6})}^{18} = 4400$ by $1100$ and simplify.

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Step 3.1

Divide each term in $1100 {(1 + \frac{x}{6})}^{18} = 4400$ by $1100$ .

$\frac{1100 {(1 + \frac{x}{6})}^{18}}{1100} = \frac{4400}{1100}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $1100$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1100 {(1 + \frac{x}{6})}^{18}}{1100} = \frac{4400}{1100}$

Step 3.2.1.2

Divide ${(1 + \frac{x}{6})}^{18}$ by $1$ .

${(1 + \frac{x}{6})}^{18} = \frac{4400}{1100}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $4400$ by $1100$ .

${(1 + \frac{x}{6})}^{18} = 4$

Step 4

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

$1 + \frac{x}{6} = \pm \sqrt[18]{4}$

Step 5

Simplify $\pm \sqrt[18]{4}$ .

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Step 5.1

Rewrite $4$ as $2^{2}$ .

$1 + \frac{x}{6} = \pm \sqrt[18]{2^{2}}$

Step 5.2

Rewrite $\sqrt[18]{2^{2}}$ as $\sqrt[9]{\sqrt{2^{2}}}$ .

$1 + \frac{x}{6} = \pm \sqrt[9]{\sqrt{2^{2}}}$

Step 5.3

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

$1 + \frac{x}{6} = \pm \sqrt[9]{2}$

Step 6

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

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Step 6.1

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

$1 + \frac{x}{6} = \sqrt[9]{2}$

Step 6.2

Subtract $1$ from both sides of the equation.

$\frac{x}{6} = \sqrt[9]{2} - 1$

Step 6.3

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 (\sqrt[9]{2} - 1)$

Step 6.4

Simplify both sides of the equation.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Cancel the common factor of $6$ .

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Step 6.4.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 (\sqrt[9]{2} - 1)$

Step 6.4.1.1.2

Rewrite the expression.

$x = 6 (\sqrt[9]{2} - 1)$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $6 (\sqrt[9]{2} - 1)$ .

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Step 6.4.2.1.1

Apply the distributive property.

$x = 6 \sqrt[9]{2} + 6 \cdot - 1$

Step 6.4.2.1.2

Multiply $6$ by $- 1$ .

$x = 6 \sqrt[9]{2} - 6$

Step 6.5

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

$1 + \frac{x}{6} = - \sqrt[9]{2}$

Step 6.6

Subtract $1$ from both sides of the equation.

$\frac{x}{6} = - \sqrt[9]{2} - 1$

Step 6.7

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 (- \sqrt[9]{2} - 1)$

Step 6.8

Simplify both sides of the equation.

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Step 6.8.1

Simplify the left side.

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Step 6.8.1.1

Cancel the common factor of $6$ .

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Step 6.8.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 (- \sqrt[9]{2} - 1)$

Step 6.8.1.1.2

Rewrite the expression.

$x = 6 (- \sqrt[9]{2} - 1)$

Step 6.8.2

Simplify the right side.

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Step 6.8.2.1

Simplify $6 (- \sqrt[9]{2} - 1)$ .

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Step 6.8.2.1.1

Apply the distributive property.

$x = 6 (- \sqrt[9]{2}) + 6 \cdot - 1$

Step 6.8.2.1.2

Multiply.

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Step 6.8.2.1.2.1

Multiply $- 1$ by $6$ .

$x = - 6 \sqrt[9]{2} + 6 \cdot - 1$

Step 6.8.2.1.2.2

Multiply $6$ by $- 1$ .

$x = - 6 \sqrt[9]{2} - 6$

Step 6.9

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

$x = 6 \sqrt[9]{2} - 6, - 6 \sqrt[9]{2} - 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 6 \sqrt[9]{2} - 6, - 6 \sqrt[9]{2} - 6$

Decimal Form:

$x = 0.48035843 \dots, - 12.48035843 \dots$