Algebra Examples

$6^{8 n + 6} = 1$ , $296$

Step 1

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

$\ln (6^{8 n + 6}) = \ln (1)$

Step 2

Expand $\ln (6^{8 n + 6})$ by moving $8 n + 6$ outside the logarithm.

$(8 n + 6) \ln (6) = \ln (1)$

Step 3

Simplify the left side.

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Apply the distributive property.

$8 n \ln (6) + 6 \ln (6) = \ln (1)$

Step 4

Simplify the right side.

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The natural logarithm of $1$ is $0$ .

$8 n \ln (6) + 6 \ln (6) = 0$

Step 5

Subtract $6 \ln (6)$ from both sides of the equation.

$8 n \ln (6) = - 6 \ln (6)$

Step 6

Divide each term in $8 n \ln (6) = - 6 \ln (6)$ by $8 \ln (6)$ and simplify.

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Divide each term in $8 n \ln (6) = - 6 \ln (6)$ by $8 \ln (6)$ .

$\frac{8 n \ln (6)}{8 \ln (6)} = \frac{- 6 \ln (6)}{8 \ln (6)}$

Simplify the left side.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$\frac{8 n \ln (6)}{8 \ln (6)} = \frac{- 6 \ln (6)}{8 \ln (6)}$

Rewrite the expression.

$\frac{n \ln (6)}{\ln (6)} = \frac{- 6 \ln (6)}{8 \ln (6)}$

Cancel the common factor of $\ln (6)$ .

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Cancel the common factor.

$\frac{n \ln (6)}{\ln (6)} = \frac{- 6 \ln (6)}{8 \ln (6)}$

Divide $n$ by $1$ .

$n = \frac{- 6 \ln (6)}{8 \ln (6)}$

Simplify the right side.

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Cancel the common factor of $- 6$ and $8$ .

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Factor $2$ out of $- 6 \ln (6)$ .

$n = \frac{2 (- 3 \ln (6))}{8 \ln (6)}$

Cancel the common factors.

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Factor $2$ out of $8 \ln (6)$ .

$n = \frac{2 (- 3 \ln (6))}{2 (4 \ln (6))}$

Cancel the common factor.

$n = \frac{2 (- 3 \ln (6))}{2 (4 \ln (6))}$

Rewrite the expression.

$n = \frac{- 3 \ln (6)}{4 \ln (6)}$

Cancel the common factor of $\ln (6)$ .

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Cancel the common factor.

$n = \frac{- 3 \ln (6)}{4 \ln (6)}$

Rewrite the expression.

$n = \frac{- 3}{4}$

Move the negative in front of the fraction.

$n = - \frac{3}{4}$

Step 7

Replace the variable $n$ with $- \frac{3}{4}$ in the expression.

$296$