Basic Math Examples

$12500 = 20000 {(1 + (\frac{0.065}{4}))}^{4 \cdot a}$

Step 1

Rewrite the equation as

$20000 {(1 + \frac{0.065}{4})}^{4 \cdot a} = 12500$ .

$20000 {(1 + \frac{0.065}{4})}^{4 \cdot a} = 12500$

Step 2

Simplify.

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Step 2.1

Divide

$0.065$ by

$4$ .

$20000 {(1 + 0.01625)}^{4 \cdot a} = 12500$

Step 2.2

Add

$1$ and

$0.01625$ .

$20000 \cdot {1.01625}^{4 \cdot a} = 12500$

Step 3

Divide each term in

$20000 \cdot {1.01625}^{4 \cdot a} = 12500$ by

$20000$ and simplify.

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Step 3.1

Divide each term in

$20000 \cdot {1.01625}^{4 \cdot a} = 12500$ by

$20000$ .

$\frac{20000 \cdot {1.01625}^{4 \cdot a}}{20000} = \frac{12500}{20000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

$20000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{20000 \cdot {1.01625}^{4 \cdot a}}{20000} = \frac{12500}{20000}$

Step 3.2.1.2

Divide

${1.01625}^{4 \cdot a}$ by

$1$ .

${1.01625}^{4 \cdot a} = \frac{12500}{20000}$

${1.01625}^{4 a} = \frac{12500}{20000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of

$12500$ and

$20000$ .

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Step 3.3.1.1

Factor

$2500$ out of

$12500$ .

${1.01625}^{4 a} = \frac{2500 (5)}{20000}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor

$2500$ out of

$20000$ .

${1.01625}^{4 a} = \frac{2500 \cdot 5}{2500 \cdot 8}$

Step 3.3.1.2.2

Cancel the common factor.

${1.01625}^{4 a} = \frac{2500 \cdot 5}{2500 \cdot 8}$

Step 3.3.1.2.3

Rewrite the expression.

${1.01625}^{4 a} = \frac{5}{8}$

Step 4

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

$\ln ({1.01625}^{4 a}) = \ln (\frac{5}{8})$

Step 5

Expand

$\ln ({1.01625}^{4 a})$ by moving

$4 a$ outside the logarithm.

$4 a \ln (1.01625) = \ln (\frac{5}{8})$

Step 6

Divide each term in

$4 a \ln (1.01625) = \ln (\frac{5}{8})$ by

$4 \ln (1.01625)$ and simplify.

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Step 6.1

Divide each term in

$4 a \ln (1.01625) = \ln (\frac{5}{8})$ by

$4 \ln (1.01625)$ .

$\frac{4 a \ln (1.01625)}{4 \ln (1.01625)} = \frac{\ln (\frac{5}{8})}{4 \ln (1.01625)}$

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.

$\frac{4 a \ln (1.01625)}{4 \ln (1.01625)} = \frac{\ln (\frac{5}{8})}{4 \ln (1.01625)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{a \ln (1.01625)}{\ln (1.01625)} = \frac{\ln (\frac{5}{8})}{4 \ln (1.01625)}$

Step 6.2.2

Cancel the common factor of

$\ln (1.01625)$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{a \ln (1.01625)}{\ln (1.01625)} = \frac{\ln (\frac{5}{8})}{4 \ln (1.01625)}$

Step 6.2.2.2

Divide

$a$ by

$1$ .

$a = \frac{\ln (\frac{5}{8})}{4 \ln (1.01625)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{\ln (\frac{5}{8})}{4 \ln (1.01625)}$

Decimal Form:

$a = - 7.28941768 \dots$