Trigonometry Examples

$9^{3 n + 2} = 81^{- 3 n}$

Step 1

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

$9^{3 n + 2} = 9^{2 (- 3 n)}$

Step 2

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

$3 n + 2 = 2 (- 3 n)$

Step 3

Solve for $n$ .

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Step 3.1

Multiply $- 3$ by $2$ .

$3 n + 2 = - 6 n$

Step 3.2

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

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Step 3.2.1

Add $6 n$ to both sides of the equation.

$3 n + 2 + 6 n = 0$

Step 3.2.2

Add $3 n$ and $6 n$ .

$9 n + 2 = 0$

Step 3.3

Subtract $2$ from both sides of the equation.

$9 n = - 2$

Step 3.4

Divide each term in $9 n = - 2$ by $9$ and simplify.

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Step 3.4.1

Divide each term in $9 n = - 2$ by $9$ .

$\frac{9 n}{9} = \frac{- 2}{9}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $9$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{9 n}{9} = \frac{- 2}{9}$

Step 3.4.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 2}{9}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$n = - \frac{2}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{2}{9}$

Decimal Form:

$n = - 0.\overline{2}$