Precalculus Examples

$\frac{1}{3} = 3^{2 x + 2}$

Step 1

Rewrite the equation as $3^{2 x + 2} = \frac{1}{3}$ .

$3^{2 x + 2} = \frac{1}{3}$

Step 2

Raise $3$ to the power of $1$ .

$3^{2 x + 2} = \frac{1}{3^{1}}$

Step 3

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{2 x + 2} = 3^{- 1}$

Step 4

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

$2 x + 2 = - 1$

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

Subtract $2$ from both sides of the equation.

$2 x = - 1 - 2$

Step 5.1.2

Subtract $2$ from $- 1$ .

$2 x = - 3$

Step 5.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$