Precalculus Examples

${(\frac{1}{9})}^{x - 5} = 3^{3 x}$

Step 1

Apply the product rule to $\frac{1}{9}$ .

$\frac{1^{x - 5}}{9^{x - 5}} = 3^{3 x}$

Step 2

One to any power is one.

$\frac{1}{9^{x - 5}} = 3^{3 x}$

Step 3

Move $9^{x - 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$9^{- (x - 5)} = 3^{3 x}$

Step 4

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

$3^{2 (- (x - 5))} = 3^{3 x}$

Step 5

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

$2 (- (x - 5)) = 3 x$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (- (x - 5))$ .

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Step 6.1.1

Rewrite.

$0 + 0 + 2 (- (x - 5)) = 3 x$

Step 6.1.2

Simplify by adding zeros.

$2 (- (x - 5)) = 3 x$

Step 6.1.3

Apply the distributive property.

$2 (- x - - 5) = 3 x$

Step 6.1.4

Multiply $- 1$ by $- 5$ .

$2 (- x + 5) = 3 x$

Step 6.1.5

Apply the distributive property.

$2 (- x) + 2 \cdot 5 = 3 x$

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $- 1$ by $2$ .

$- 2 x + 2 \cdot 5 = 3 x$

Step 6.1.6.2

Multiply $2$ by $5$ .

$- 2 x + 10 = 3 x$

Step 6.2

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

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Step 6.2.1

Subtract $3 x$ from both sides of the equation.

$- 2 x + 10 - 3 x = 0$

Step 6.2.2

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

$- 5 x + 10 = 0$

Step 6.3

Subtract $10$ from both sides of the equation.

$- 5 x = - 10$

Step 6.4

Divide each term in $- 5 x = - 10$ by $- 5$ and simplify.

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Step 6.4.1

Divide each term in $- 5 x = - 10$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 10}{- 5}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $- 5$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 10}{- 5}$

Step 6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{- 5}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Divide $- 10$ by $- 5$ .

$x = 2$