Basic Math Examples

$27^{3 n} = 81^{- n}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$3^{3 (3 n)} = 3^{4 (- n)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 (3 n) = 4 (- n)$

Step 3

Solve for $n$ .

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Step 3.1

Multiply $3$ by $3$ .

$9 n = 4 (- n)$

Step 3.2

Multiply $- 1$ by $4$ .

$9 n = - 4 n$

Step 3.3

Move all terms containing $n$ to the left side of the equation.

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Step 3.3.1

Add $4 n$ to both sides of the equation.

$9 n + 4 n = 0$

Step 3.3.2

Add $9 n$ and $4 n$ .

$13 n = 0$

Step 3.4

Divide each term in $13 n = 0$ by $13$ and simplify.

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Step 3.4.1

Divide each term in $13 n = 0$ by $13$ .

$\frac{13 n}{13} = \frac{0}{13}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $13$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{13 n}{13} = \frac{0}{13}$

Step 3.4.2.1.2

Divide $n$ by $1$ .

$n = \frac{0}{13}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $13$ .

$n = 0$