Basic Math Examples

$1 - (136 \cdot 1 - 0.00000836 {(1 + r)}^{- 136}) = 0.07441$

Step 1

Simplify $1 - (136 \cdot 1 - 0.00000836 {(1 + r)}^{- 136})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Multiply $136$ by $1$ .

$1 - (136 - 0.00000836 {(1 + r)}^{- 136}) = 0.07441$

Step 1.1.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - (136 - 0.00000836 \frac{1}{{(1 + r)}^{136}}) = 0.07441$

Step 1.1.1.3

Combine $- 0.00000836$ and $\frac{1}{{(1 + r)}^{136}}$ .

$1 - (136 + \frac{- 0.00000836}{{(1 + r)}^{136}}) = 0.07441$

Step 1.1.1.4

Move the negative in front of the fraction.

$1 - (136 - \frac{0.00000836}{{(1 + r)}^{136}}) = 0.07441$

Step 1.1.2

Apply the distributive property.

$1 - 1 \cdot 136 - - \frac{0.00000836}{{(1 + r)}^{136}} = 0.07441$

Step 1.1.3

Multiply $- 1$ by $136$ .

$1 - 136 - - \frac{0.00000836}{{(1 + r)}^{136}} = 0.07441$

Step 1.1.4

Multiply $- - \frac{0.00000836}{{(1 + r)}^{136}}$ .

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Step 1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 - 136 + 1 \frac{0.00000836}{{(1 + r)}^{136}} = 0.07441$

Step 1.1.4.2

Multiply $\frac{0.00000836}{{(1 + r)}^{136}}$ by $1$ .

$1 - 136 + \frac{0.00000836}{{(1 + r)}^{136}} = 0.07441$

Step 1.2

Subtract $136$ from $1$ .

$- 135 + \frac{0.00000836}{{(1 + r)}^{136}} = 0.07441$

Step 2

Move all terms not containing $r$ to the right side of the equation.

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Step 2.1

Add $135$ to both sides of the equation.

$\frac{0.00000836}{{(1 + r)}^{136}} = 0.07441 + 135$

Step 2.2

Add $0.07441$ and $135$ .

$\frac{0.00000836}{{(1 + r)}^{136}} = 135.07441$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

${(1 + r)}^{136}, 1$

Step 3.2

The LCM of one and any expression is the expression.

${(1 + r)}^{136}$

Step 4

Multiply each term in $\frac{0.00000836}{{(1 + r)}^{136}} = 135.07441$ by ${(1 + r)}^{136}$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{0.00000836}{{(1 + r)}^{136}} = 135.07441$ by ${(1 + r)}^{136}$ .

$\frac{0.00000836}{{(1 + r)}^{136}} {(1 + r)}^{136} = 135.07441 {(1 + r)}^{136}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of ${(1 + r)}^{136}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.00000836}{{(1 + r)}^{136}} {(1 + r)}^{136} = 135.07441 {(1 + r)}^{136}$

Step 4.2.1.2

Rewrite the expression.

$0.00000836 = 135.07441 {(1 + r)}^{136}$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $135.07441 {(1 + r)}^{136} = 0.00000836$ .

$135.07441 {(1 + r)}^{136} = 0.00000836$

Step 5.2

Divide each term in $135.07441 {(1 + r)}^{136} = 0.00000836$ by $135.07441$ and simplify.

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Step 5.2.1

Divide each term in $135.07441 {(1 + r)}^{136} = 0.00000836$ by $135.07441$ .

$\frac{135.07441 {(1 + r)}^{136}}{135.07441} = \frac{0.00000836}{135.07441}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $135.07441$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{135.07441 {(1 + r)}^{136}}{135.07441} = \frac{0.00000836}{135.07441}$

Step 5.2.2.1.2

Divide ${(1 + r)}^{136}$ by $1$ .

${(1 + r)}^{136} = \frac{0.00000836}{135.07441}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0.00000836$ by $135.07441$ .

${(1 + r)}^{136} = 0.00000006$

Step 5.3

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

$1 + r = \pm \sqrt[136]{0.00000006}$

Step 5.4

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

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Step 5.4.1

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

$1 + r = \sqrt[136]{0.00000006}$

Step 5.4.2

Subtract $1$ from both sides of the equation.

$r = \sqrt[136]{0.00000006} - 1$

Step 5.4.3

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

$1 + r = - \sqrt[136]{0.00000006}$

Step 5.4.4

Subtract $1$ from both sides of the equation.

$r = - \sqrt[136]{0.00000006} - 1$

Step 5.4.5

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

$r = \sqrt[136]{0.00000006} - 1, - \sqrt[136]{0.00000006} - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \sqrt[136]{0.00000006} - 1, - \sqrt[136]{0.00000006} - 1$

Decimal Form:

$r = - 0.11488988 \dots, - 1.88511011 \dots$