Finite Math Examples

${(\frac{1 + x}{1 + \frac{x}{2}})}^{20} = 1.4$

Step 1

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

$\frac{1 + x}{1 + \frac{x}{2}} = \pm \sqrt[20]{1.4}$

Step 2

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

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Step 2.1

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

$\frac{1 + x}{1 + \frac{x}{2}} = \sqrt[20]{1.4}$

Step 2.2

Factor each term.

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Step 2.2.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 2.2.1.1

Multiply $\frac{1 + x}{1 + \frac{x}{2}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{1 + x}{1 + \frac{x}{2}} = \sqrt[20]{1.4}$

Step 2.2.1.2

Combine.

$\frac{2 (1 + x)}{2 (1 + \frac{x}{2})} = \sqrt[20]{1.4}$

Step 2.2.2

Apply the distributive property.

$\frac{2 \cdot 1 + 2 x}{2 \cdot 1 + 2 \frac{x}{2}} = \sqrt[20]{1.4}$

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Cancel the common factor.

$\frac{2 \cdot 1 + 2 x}{2 \cdot 1 + 2 \frac{x}{2}} = \sqrt[20]{1.4}$

Step 2.2.3.2

Rewrite the expression.

$\frac{2 \cdot 1 + 2 x}{2 \cdot 1 + x} = \sqrt[20]{1.4}$

Step 2.2.4

Factor $2$ out of $2 \cdot 1 + 2 x$ .

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Step 2.2.4.1

Factor $2$ out of $2 \cdot 1$ .

$\frac{2 (1) + 2 x}{2 \cdot 1 + x} = \sqrt[20]{1.4}$

Step 2.2.4.2

Factor $2$ out of $2 x$ .

$\frac{2 (1) + 2 (x)}{2 \cdot 1 + x} = \sqrt[20]{1.4}$

Step 2.2.4.3

Factor $2$ out of $2 (1) + 2 (x)$ .

$\frac{2 (1 + x)}{2 \cdot 1 + x} = \sqrt[20]{1.4}$

Step 2.2.5

Multiply $2$ by $1$ .

$\frac{2 (1 + x)}{2 + x} = \sqrt[20]{1.4}$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 + x, 1$

Step 2.3.2

Remove parentheses.

$2 + x, 1$

Step 2.3.3

The LCM of one and any expression is the expression.

$2 + x$

Step 2.4

Multiply each term in $\frac{2 (1 + x)}{2 + x} = \sqrt[20]{1.4}$ by $2 + x$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{2 (1 + x)}{2 + x} = \sqrt[20]{1.4}$ by $2 + x$ .

$\frac{2 (1 + x)}{2 + x} (2 + x) = \sqrt[20]{1.4} (2 + x)$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2 + x$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{2 (1 + x)}{2 + x} (2 + x) = \sqrt[20]{1.4} (2 + x)$

Step 2.4.2.1.2

Rewrite the expression.

$2 (1 + x) = \sqrt[20]{1.4} (2 + x)$

Step 2.4.2.2

Apply the distributive property.

$2 \cdot 1 + 2 x = \sqrt[20]{1.4} (2 + x)$

Step 2.4.2.3

Multiply $2$ by $1$ .

$2 + 2 x = \sqrt[20]{1.4} (2 + x)$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Apply the distributive property.

$2 + 2 x = \sqrt[20]{1.4} \cdot 2 + \sqrt[20]{1.4} x$

Step 2.4.3.2

Move $2$ to the left of $\sqrt[20]{1.4}$ .

$2 + 2 x = 2 \sqrt[20]{1.4} + \sqrt[20]{1.4} x$

Step 2.5

Solve the equation.

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Step 2.5.1

Subtract $\sqrt[20]{1.4} x$ from both sides of the equation.

$2 + 2 x - \sqrt[20]{1.4} x = 2 \sqrt[20]{1.4}$

Step 2.5.2

Subtract $2$ from both sides of the equation.

$2 x - \sqrt[20]{1.4} x = 2 \sqrt[20]{1.4} - 2$

Step 2.5.3

Factor $x$ out of $2 x - \sqrt[20]{1.4} x$ .

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Step 2.5.3.1

Factor $x$ out of $2 x$ .

$x \cdot 2 - \sqrt[20]{1.4} x = 2 \sqrt[20]{1.4} - 2$

Step 2.5.3.2

Factor $x$ out of $- \sqrt[20]{1.4} x$ .

$x \cdot 2 + x (- \sqrt[20]{1.4}) = 2 \sqrt[20]{1.4} - 2$

Step 2.5.3.3

Factor $x$ out of $x \cdot 2 + x (- \sqrt[20]{1.4})$ .

$x (2 - \sqrt[20]{1.4}) = 2 \sqrt[20]{1.4} - 2$

Step 2.5.4

Divide each term in $x (2 - \sqrt[20]{1.4}) = 2 \sqrt[20]{1.4} - 2$ by $2 - \sqrt[20]{1.4}$ and simplify.

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Step 2.5.4.1

Divide each term in $x (2 - \sqrt[20]{1.4}) = 2 \sqrt[20]{1.4} - 2$ by $2 - \sqrt[20]{1.4}$ .

$\frac{x (2 - \sqrt[20]{1.4})}{2 - \sqrt[20]{1.4}} = \frac{2 \sqrt[20]{1.4}}{2 - \sqrt[20]{1.4}} + \frac{- 2}{2 - \sqrt[20]{1.4}}$

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $2 - \sqrt[20]{1.4}$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

$\frac{x (2 - \sqrt[20]{1.4})}{2 - \sqrt[20]{1.4}} = \frac{2 \sqrt[20]{1.4}}{2 - \sqrt[20]{1.4}} + \frac{- 2}{2 - \sqrt[20]{1.4}}$

Step 2.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 \sqrt[20]{1.4}}{2 - \sqrt[20]{1.4}} + \frac{- 2}{2 - \sqrt[20]{1.4}}$

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Combine the numerators over the common denominator.

$x = \frac{2 \sqrt[20]{1.4} - 2}{2 - \sqrt[20]{1.4}}$

Step 2.5.4.3.2

Factor $2$ out of $2 \sqrt[20]{1.4} - 2$ .

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Step 2.5.4.3.2.1

Factor $2$ out of $2 \sqrt[20]{1.4}$ .

$x = \frac{2 \sqrt[20]{1.4} - 2}{2 - \sqrt[20]{1.4}}$

Step 2.5.4.3.2.2

Factor $2$ out of $- 2$ .

$x = \frac{2 \sqrt[20]{1.4} + 2 (- 1)}{2 - \sqrt[20]{1.4}}$

Step 2.5.4.3.2.3

Factor $2$ out of $2 \sqrt[20]{1.4} + 2 (- 1)$ .

$x = \frac{2 (\sqrt[20]{1.4} - 1)}{2 - \sqrt[20]{1.4}}$

Step 2.6

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

$\frac{1 + x}{1 + \frac{x}{2}} = - \sqrt[20]{1.4}$

Step 2.7

Factor each term.

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Step 2.7.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 2.7.1.1

Multiply $\frac{1 + x}{1 + \frac{x}{2}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{1 + x}{1 + \frac{x}{2}} = - \sqrt[20]{1.4}$

Step 2.7.1.2

Combine.

$\frac{2 (1 + x)}{2 (1 + \frac{x}{2})} = - \sqrt[20]{1.4}$

Step 2.7.2

Apply the distributive property.

$\frac{2 \cdot 1 + 2 x}{2 \cdot 1 + 2 \frac{x}{2}} = - \sqrt[20]{1.4}$

Step 2.7.3

Cancel the common factor of $2$ .

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Step 2.7.3.1

Cancel the common factor.

$\frac{2 \cdot 1 + 2 x}{2 \cdot 1 + 2 \frac{x}{2}} = - \sqrt[20]{1.4}$

Step 2.7.3.2

Rewrite the expression.

$\frac{2 \cdot 1 + 2 x}{2 \cdot 1 + x} = - \sqrt[20]{1.4}$

Step 2.7.4

Factor $2$ out of $2 \cdot 1 + 2 x$ .

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Step 2.7.4.1

Factor $2$ out of $2 \cdot 1$ .

$\frac{2 (1) + 2 x}{2 \cdot 1 + x} = - \sqrt[20]{1.4}$

Step 2.7.4.2

Factor $2$ out of $2 x$ .

$\frac{2 (1) + 2 (x)}{2 \cdot 1 + x} = - \sqrt[20]{1.4}$

Step 2.7.4.3

Factor $2$ out of $2 (1) + 2 (x)$ .

$\frac{2 (1 + x)}{2 \cdot 1 + x} = - \sqrt[20]{1.4}$

Step 2.7.5

Multiply $2$ by $1$ .

$\frac{2 (1 + x)}{2 + x} = - \sqrt[20]{1.4}$

Step 2.8

Find the LCD of the terms in the equation.

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Step 2.8.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 + x, 1$

Step 2.8.2

Remove parentheses.

$2 + x, 1$

Step 2.8.3

The LCM of one and any expression is the expression.

$2 + x$

Step 2.9

Multiply each term in $\frac{2 (1 + x)}{2 + x} = - \sqrt[20]{1.4}$ by $2 + x$ to eliminate the fractions.

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Step 2.9.1

Multiply each term in $\frac{2 (1 + x)}{2 + x} = - \sqrt[20]{1.4}$ by $2 + x$ .

$\frac{2 (1 + x)}{2 + x} (2 + x) = - \sqrt[20]{1.4} (2 + x)$

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $2 + x$ .

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Step 2.9.2.1.1

Cancel the common factor.

$\frac{2 (1 + x)}{2 + x} (2 + x) = - \sqrt[20]{1.4} (2 + x)$

Step 2.9.2.1.2

Rewrite the expression.

$2 (1 + x) = - \sqrt[20]{1.4} (2 + x)$

Step 2.9.2.2

Apply the distributive property.

$2 \cdot 1 + 2 x = - \sqrt[20]{1.4} (2 + x)$

Step 2.9.2.3

Multiply $2$ by $1$ .

$2 + 2 x = - \sqrt[20]{1.4} (2 + x)$

Step 2.9.3

Simplify the right side.

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Step 2.9.3.1

Apply the distributive property.

$2 + 2 x = - \sqrt[20]{1.4} \cdot 2 - \sqrt[20]{1.4} x$

Step 2.9.3.2

Multiply $2$ by $- 1$ .

$2 + 2 x = - 2 \sqrt[20]{1.4} - \sqrt[20]{1.4} x$

Step 2.10

Solve the equation.

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Step 2.10.1

Add $\sqrt[20]{1.4} x$ to both sides of the equation.

$2 + 2 x + \sqrt[20]{1.4} x = - 2 \sqrt[20]{1.4}$

Step 2.10.2

Subtract $2$ from both sides of the equation.

$2 x + \sqrt[20]{1.4} x = - 2 \sqrt[20]{1.4} - 2$

Step 2.10.3

Factor $x$ out of $2 x + \sqrt[20]{1.4} x$ .

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Step 2.10.3.1

Factor $x$ out of $2 x$ .

$x \cdot 2 + \sqrt[20]{1.4} x = - 2 \sqrt[20]{1.4} - 2$

Step 2.10.3.2

Factor $x$ out of $\sqrt[20]{1.4} x$ .

$x \cdot 2 + x \sqrt[20]{1.4} = - 2 \sqrt[20]{1.4} - 2$

Step 2.10.3.3

Factor $x$ out of $x \cdot 2 + x \sqrt[20]{1.4}$ .

$x (2 + \sqrt[20]{1.4}) = - 2 \sqrt[20]{1.4} - 2$

Step 2.10.4

Divide each term in $x (2 + \sqrt[20]{1.4}) = - 2 \sqrt[20]{1.4} - 2$ by $2 + \sqrt[20]{1.4}$ and simplify.

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Step 2.10.4.1

Divide each term in $x (2 + \sqrt[20]{1.4}) = - 2 \sqrt[20]{1.4} - 2$ by $2 + \sqrt[20]{1.4}$ .

$\frac{x (2 + \sqrt[20]{1.4})}{2 + \sqrt[20]{1.4}} = \frac{- 2 \sqrt[20]{1.4}}{2 + \sqrt[20]{1.4}} + \frac{- 2}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.2

Simplify the left side.

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Step 2.10.4.2.1

Cancel the common factor of $2 + \sqrt[20]{1.4}$ .

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Step 2.10.4.2.1.1

Cancel the common factor.

$\frac{x (2 + \sqrt[20]{1.4})}{2 + \sqrt[20]{1.4}} = \frac{- 2 \sqrt[20]{1.4}}{2 + \sqrt[20]{1.4}} + \frac{- 2}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 \sqrt[20]{1.4}}{2 + \sqrt[20]{1.4}} + \frac{- 2}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3

Simplify the right side.

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Step 2.10.4.3.1

Combine the numerators over the common denominator.

$x = \frac{- 2 \sqrt[20]{1.4} - 2}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.2

Factor $2$ out of $- 2 \sqrt[20]{1.4} - 2$ .

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Step 2.10.4.3.2.1

Factor $2$ out of $- 2 \sqrt[20]{1.4}$ .

$x = \frac{2 (- \sqrt[20]{1.4}) - 2}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.2.2

Factor $2$ out of $- 2$ .

$x = \frac{2 (- \sqrt[20]{1.4}) + 2 (- 1)}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.2.3

Factor $2$ out of $2 (- \sqrt[20]{1.4}) + 2 (- 1)$ .

$x = \frac{2 (- \sqrt[20]{1.4} - 1)}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.3

Factor $- 1$ out of $- \sqrt[20]{1.4}$ .

$x = \frac{2 (- \sqrt[20]{1.4} - 1)}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{2 (- \sqrt[20]{1.4} - 1 (1))}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.5

Factor $- 1$ out of $- \sqrt[20]{1.4} - 1 (1)$ .

$x = \frac{2 (- (\sqrt[20]{1.4} + 1))}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.6

Simplify the expression.

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Step 2.10.4.3.6.1

Rewrite $- (\sqrt[20]{1.4} + 1)$ as $- 1 (\sqrt[20]{1.4} + 1)$ .

$x = \frac{2 (- 1 (\sqrt[20]{1.4} + 1))}{2 + \sqrt[20]{1.4}}$

Step 2.10.4.3.6.2

Move the negative in front of the fraction.

$x = - \frac{2 (\sqrt[20]{1.4} + 1)}{2 + \sqrt[20]{1.4}}$

Step 2.11

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

$x = \frac{2 (\sqrt[20]{1.4} - 1)}{2 - \sqrt[20]{1.4}}, - \frac{2 (\sqrt[20]{1.4} + 1)}{2 + \sqrt[20]{1.4}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 (\sqrt[20]{1.4} - 1)}{2 - \sqrt[20]{1.4}}, - \frac{2 (\sqrt[20]{1.4} + 1)}{2 + \sqrt[20]{1.4}}$

Decimal Form:

$x = 0.03451747 \dots, - 1.33708233 \dots$