Basic Math Examples

$18 = \frac{1}{12^{n^{2} + 200}}$

Step 1

Rewrite the equation as $\frac{1}{12^{n^{2} + 200}} = 18$ .

$\frac{1}{12^{n^{2} + 200}} = 18$

Step 2

Multiply both sides by $12^{n^{2} + 200}$ .

$\frac{1}{12^{n^{2} + 200}} \cdot 12^{n^{2} + 200} = 18 \cdot 12^{n^{2} + 200}$

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $12^{n^{2} + 200}$ .

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Step 3.1.1

Cancel the common factor.

$\frac{1}{12^{n^{2} + 200}} \cdot 12^{n^{2} + 200} = 18 \cdot 12^{n^{2} + 200}$

Step 3.1.2

Rewrite the expression.

$1 = 18 \cdot 12^{n^{2} + 200}$

Step 4

Solve for $n$ .

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Step 4.1

Rewrite the equation as $18 \cdot 12^{n^{2} + 200} = 1$ .

$18 \cdot 12^{n^{2} + 200} = 1$

Step 4.2

Divide each term in $18 \cdot 12^{n^{2} + 200} = 1$ by $18$ and simplify.

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Step 4.2.1

Divide each term in $18 \cdot 12^{n^{2} + 200} = 1$ by $18$ .

$\frac{18 \cdot 12^{n^{2} + 200}}{18} = \frac{1}{18}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $18$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{18 \cdot 12^{n^{2} + 200}}{18} = \frac{1}{18}$

Step 4.2.2.1.2

Divide $12^{n^{2} + 200}$ by $1$ .

$12^{n^{2} + 200} = \frac{1}{18}$

Step 4.3

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

$\ln (12^{n^{2} + 200}) = \ln (\frac{1}{18})$

Step 4.4

Expand $\ln (12^{n^{2} + 200})$ by moving $n^{2} + 200$ outside the logarithm.

$(n^{2} + 200) \ln (12) = \ln (\frac{1}{18})$

Step 4.5

Simplify the left side.

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Step 4.5.1

Apply the distributive property.

$n^{2} \ln (12) + 200 \ln (12) = \ln (\frac{1}{18})$

Step 4.6

Move all the terms containing a logarithm to the left side of the equation.

$n^{2} \ln (12) + 200 \ln (12) - \ln (\frac{1}{18}) = 0$

Step 4.7

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

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Step 4.7.1

Subtract $200 \ln (12)$ from both sides of the equation.

$n^{2} \ln (12) - \ln (\frac{1}{18}) = - 200 \ln (12)$

Step 4.7.2

Add $\ln (\frac{1}{18})$ to both sides of the equation.

$n^{2} \ln (12) = - 200 \ln (12) + \ln (\frac{1}{18})$

Step 4.8

Divide each term in $n^{2} \ln (12) = - 200 \ln (12) + \ln (\frac{1}{18})$ by $\ln (12)$ and simplify.

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Step 4.8.1

Divide each term in $n^{2} \ln (12) = - 200 \ln (12) + \ln (\frac{1}{18})$ by $\ln (12)$ .

$\frac{n^{2} \ln (12)}{\ln (12)} = \frac{- 200 \ln (12)}{\ln (12)} + \frac{\ln (\frac{1}{18})}{\ln (12)}$

Step 4.8.2

Simplify the left side.

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Step 4.8.2.1

Cancel the common factor of $\ln (12)$ .

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Step 4.8.2.1.1

Cancel the common factor.

$\frac{n^{2} \ln (12)}{\ln (12)} = \frac{- 200 \ln (12)}{\ln (12)} + \frac{\ln (\frac{1}{18})}{\ln (12)}$

Step 4.8.2.1.2

Divide $n^{2}$ by $1$ .

$n^{2} = \frac{- 200 \ln (12)}{\ln (12)} + \frac{\ln (\frac{1}{18})}{\ln (12)}$

Step 4.8.3

Simplify the right side.

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Step 4.8.3.1

Cancel the common factor of $\ln (12)$ .

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Step 4.8.3.1.1

Cancel the common factor.

$n^{2} = \frac{- 200 \ln (12)}{\ln (12)} + \frac{\ln (\frac{1}{18})}{\ln (12)}$

Step 4.8.3.1.2

Divide $- 200$ by $1$ .

$n^{2} = - 200 + \frac{\ln (\frac{1}{18})}{\ln (12)}$

Step 4.9

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

$n = \pm \sqrt{- 200 + \frac{\ln (\frac{1}{18})}{\ln (12)}}$

Step 4.10

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

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Step 4.10.1

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

$n = \sqrt{- 200 + \frac{\ln (\frac{1}{18})}{\ln (12)}}$

Step 4.10.2

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

$n = - \sqrt{- 200 + \frac{\ln (\frac{1}{18})}{\ln (12)}}$

Step 4.10.3

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

$n = \sqrt{- 200 + \frac{\ln (\frac{1}{18})}{\ln (12)}}, - \sqrt{- 200 + \frac{\ln (\frac{1}{18})}{\ln (12)}}$

Step 5

Exclude the solutions that do not make $18 = \frac{1}{12^{n^{2} + 200}}$ true.

$No solution$