Precalculus Examples

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

Step 1

Apply the product rule to $\frac{1}{3}$ .

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

Step 2

One to any power is one.

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

Step 3

Move $3^{3 x + 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{- (3 x + 6)} = 9^{x - 3}$

Step 4

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

$3^{- (3 x + 6)} = 3^{2 (x - 3)}$

Step 5

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

$- (3 x + 6) = 2 (x - 3)$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $- (3 x + 6)$ .

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Step 6.1.1

Rewrite.

$0 + 0 - (3 x + 6) = 2 (x - 3)$

Step 6.1.2

Simplify by adding zeros.

$- (3 x + 6) = 2 (x - 3)$

Step 6.1.3

Apply the distributive property.

$- (3 x) - 1 \cdot 6 = 2 (x - 3)$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $3$ by $- 1$ .

$- 3 x - 1 \cdot 6 = 2 (x - 3)$

Step 6.1.4.2

Multiply $- 1$ by $6$ .

$- 3 x - 6 = 2 (x - 3)$

Step 6.2

Simplify $2 (x - 3)$ .

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Step 6.2.1

Apply the distributive property.

$- 3 x - 6 = 2 x + 2 \cdot - 3$

Step 6.2.2

Multiply $2$ by $- 3$ .

$- 3 x - 6 = 2 x - 6$

Step 6.3

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

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Step 6.3.1

Subtract $2 x$ from both sides of the equation.

$- 3 x - 6 - 2 x = - 6$

Step 6.3.2

Subtract $2 x$ from $- 3 x$ .

$- 5 x - 6 = - 6$

Step 6.4

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

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Step 6.4.1

Add $6$ to both sides of the equation.

$- 5 x = - 6 + 6$

Step 6.4.2

Add $- 6$ and $6$ .

$- 5 x = 0$

Step 6.5

Divide each term in $- 5 x = 0$ by $- 5$ and simplify.

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Step 6.5.1

Divide each term in $- 5 x = 0$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{0}{- 5}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 5$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{0}{- 5}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 5}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $0$ by $- 5$ .

$x = 0$