Precalculus Examples

$9^{x} = \frac{3}{3^{x}}$

Step 1

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

$9^{x} = 3 \cdot 3^{- x}$

Step 2

Multiply $3$ by $3^{- x}$ .

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Step 2.1

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

$9^{x} = 3^{1} \cdot 3^{- x}$

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3

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

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

Step 4

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

$2 x = 1 - x$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Add $x$ to both sides of the equation.

$2 x + x = 1$

Step 5.1.2

Add $2 x$ and $x$ .

$3 x = 1$

Step 5.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = 1$ by $3$ .

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

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.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}$