Basic Math Examples

$56 = \frac{1 - {(1 + 0.015)}^{- n}}{0.015}$

Step 1

Rewrite the equation as $\frac{1 - {(1 + 0.015)}^{- n}}{0.015} = 56$ .

$\frac{1 - {(1 + 0.015)}^{- n}}{0.015} = 56$

Step 2

Multiply both sides by $0.015$ .

$\frac{1 - {(1 + 0.015)}^{- n}}{0.015} \cdot 0.015 = 56 \cdot 0.015$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{1 - {(1 + 0.015)}^{- n}}{0.015} \cdot 0.015$ .

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Step 3.1.1.1

Add $1$ and $0.015$ .

$\frac{1 - {1.015}^{- n}}{0.015} \cdot 0.015 = 56 \cdot 0.015$

Step 3.1.1.2

Cancel the common factor of $0.015$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{1 - {1.015}^{- n}}{0.015} \cdot 0.015 = 56 \cdot 0.015$

Step 3.1.1.2.2

Rewrite the expression.

$1 - {1.015}^{- n} = 56 \cdot 0.015$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $56$ by $0.015$ .

$1 - {1.015}^{- n} = 0.84$

Step 4

Solve for $n$ .

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Step 4.1

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

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Step 4.1.1

Subtract $1$ from both sides of the equation.

$- {1.015}^{- n} = 0.84 - 1$

Step 4.1.2

Subtract $1$ from $0.84$ .

$- {1.015}^{- n} = - 0.16$

Step 4.2

Divide each term in $- {1.015}^{- n} = - 0.16$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- {1.015}^{- n} = - 0.16$ by $- 1$ .

$\frac{- {1.015}^{- n}}{- 1} = \frac{- 0.16}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{{1.015}^{- n}}{1} = \frac{- 0.16}{- 1}$

Step 4.2.2.2

Divide ${1.015}^{- n}$ by $1$ .

${1.015}^{- n} = \frac{- 0.16}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 0.16$ by $- 1$ .

${1.015}^{- n} = 0.16$

Step 4.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({1.015}^{- n}) = \ln (0.16)$

Step 4.4

Expand $\ln ({1.015}^{- n})$ by moving $- n$ outside the logarithm.

$- n \ln (1.015) = \ln (0.16)$

Step 4.5

Divide each term in $- n \ln (1.015) = \ln (0.16)$ by $- \ln (1.015)$ and simplify.

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Step 4.5.1

Divide each term in $- n \ln (1.015) = \ln (0.16)$ by $- \ln (1.015)$ .

$\frac{- n \ln (1.015)}{- \ln (1.015)} = \frac{\ln (0.16)}{- \ln (1.015)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Dividing two negative values results in a positive value.

$\frac{n \ln (1.015)}{\ln (1.015)} = \frac{\ln (0.16)}{- \ln (1.015)}$

Step 4.5.2.2

Cancel the common factor of $\ln (1.015)$ .

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Step 4.5.2.2.1

Cancel the common factor.

$\frac{n \ln (1.015)}{\ln (1.015)} = \frac{\ln (0.16)}{- \ln (1.015)}$

Step 4.5.2.2.2

Divide $n$ by $1$ .

$n = \frac{\ln (0.16)}{- \ln (1.015)}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (0.16)}{\ln (1.015)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (0.16)}{\ln (1.015)}$

Decimal Form:

$n = 123.08611460 \dots$