Precalculus Examples

$3^{3 x - 6} = \frac{1}{9}$

Step 1

Raise $9$ to the power of $1$ .

$3^{3 x - 6} = \frac{1}{9^{1}}$

Step 2

Move $9^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{3 x - 6} = 9^{- 1}$

Step 3

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

$3^{3 x - 6} = 3^{- 2}$

Step 4

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

$3 x - 6 = - 2$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Add $6$ to both sides of the equation.

$3 x = - 2 + 6$

Step 5.1.2

Add $- 2$ and $6$ .

$3 x = 4$

Step 5.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}$

Decimal Form:

$x = 1.\overline{3}$

Mixed Number Form:

$x = 1 \frac{1}{3}$