Basic Math Examples

$18000 = 10000 {(1 + \frac{0.04}{4})}^{4 n}$

Step 1

Rewrite the equation as

$10000 {(1 + \frac{0.04}{4})}^{4 n} = 18000$ .

$10000 {(1 + \frac{0.04}{4})}^{4 n} = 18000$

Step 2

Simplify.

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Step 2.1

Divide

$0.04$ by

$4$ .

$10000 {(1 + 0.01)}^{4 n} = 18000$

Step 2.2

Add

$1$ and

$0.01$ .

$10000 \cdot {1.01}^{4 n} = 18000$

Step 3

Divide each term in

$10000 \cdot {1.01}^{4 n} = 18000$ by

$10000$ and simplify.

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Step 3.1

Divide each term in

$10000 \cdot {1.01}^{4 n} = 18000$ by

$10000$ .

$\frac{10000 \cdot {1.01}^{4 n}}{10000} = \frac{18000}{10000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

$10000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{10000 \cdot {1.01}^{4 n}}{10000} = \frac{18000}{10000}$

Step 3.2.1.2

Divide

${1.01}^{4 n}$ by

$1$ .

${1.01}^{4 n} = \frac{18000}{10000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of

$18000$ and

$10000$ .

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Step 3.3.1.1

Factor

$2000$ out of

$18000$ .

${1.01}^{4 n} = \frac{2000 (9)}{10000}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor

$2000$ out of

$10000$ .

${1.01}^{4 n} = \frac{2000 \cdot 9}{2000 \cdot 5}$

Step 3.3.1.2.2

Cancel the common factor.

${1.01}^{4 n} = \frac{2000 \cdot 9}{2000 \cdot 5}$

Step 3.3.1.2.3

Rewrite the expression.

${1.01}^{4 n} = \frac{9}{5}$

Step 4

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

$\ln ({1.01}^{4 n}) = \ln (\frac{9}{5})$

Step 5

Expand

$\ln ({1.01}^{4 n})$ by moving

$4 n$ outside the logarithm.

$4 n \ln (1.01) = \ln (\frac{9}{5})$

Step 6

Divide each term in

$4 n \ln (1.01) = \ln (\frac{9}{5})$ by

$4 \ln (1.01)$ and simplify.

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Step 6.1

Divide each term in

$4 n \ln (1.01) = \ln (\frac{9}{5})$ by

$4 \ln (1.01)$ .

$\frac{4 n \ln (1.01)}{4 \ln (1.01)} = \frac{\ln (\frac{9}{5})}{4 \ln (1.01)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

$4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{4 n \ln (1.01)}{4 \ln (1.01)} = \frac{\ln (\frac{9}{5})}{4 \ln (1.01)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{n \ln (1.01)}{\ln (1.01)} = \frac{\ln (\frac{9}{5})}{4 \ln (1.01)}$

Step 6.2.2

Cancel the common factor of

$\ln (1.01)$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{n \ln (1.01)}{\ln (1.01)} = \frac{\ln (\frac{9}{5})}{4 \ln (1.01)}$

Step 6.2.2.2

Divide

$n$ by

$1$ .

$n = \frac{\ln (\frac{9}{5})}{4 \ln (1.01)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (\frac{9}{5})}{4 \ln (1.01)}$

Decimal Form:

$n = 14.76801810 \dots$