Finite Math Examples

$9^{x + 2} = {(\frac{1}{27})}^{x + 12}$

Step 1

Apply the product rule to $\frac{1}{27}$ .

$9^{x + 2} = \frac{1^{x + 12}}{27^{x + 12}}$

Step 2

One to any power is one.

$9^{x + 2} = \frac{1}{27^{x + 12}}$

Step 3

Move $27^{x + 12}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$9^{x + 2} = 27^{- (x + 12)}$

Step 4

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

$3^{2 (x + 2)} = 3^{3 (- (x + 12))}$

Step 5

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

$2 (x + 2) = 3 (- (x + 12))$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify $2 (x + 2)$ .

Tap for more steps...

Step 6.1.1

Rewrite.

$0 + 0 + 2 (x + 2) = 3 (- (x + 12))$

Step 6.1.2

Simplify by adding zeros.

$2 (x + 2) = 3 (- (x + 12))$

Step 6.1.3

Apply the distributive property.

$2 x + 2 \cdot 2 = 3 (- (x + 12))$

Step 6.1.4

Multiply $2$ by $2$ .

$2 x + 4 = 3 (- (x + 12))$

Step 6.2

Simplify $3 (- (x + 12))$ .

Tap for more steps...

Step 6.2.1

Apply the distributive property.

$2 x + 4 = 3 (- x - 1 \cdot 12)$

Step 6.2.2

Multiply $- 1$ by $12$ .

$2 x + 4 = 3 (- x - 12)$

Step 6.2.3

Apply the distributive property.

$2 x + 4 = 3 (- x) + 3 \cdot - 12$

Step 6.2.4

Multiply.

Tap for more steps...

Step 6.2.4.1

Multiply $- 1$ by $3$ .

$2 x + 4 = - 3 x + 3 \cdot - 12$

Step 6.2.4.2

Multiply $3$ by $- 12$ .

$2 x + 4 = - 3 x - 36$

Step 6.3

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

Tap for more steps...

Step 6.3.1

Add $3 x$ to both sides of the equation.

$2 x + 4 + 3 x = - 36$

Step 6.3.2

Add $2 x$ and $3 x$ .

$5 x + 4 = - 36$

Step 6.4

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

Tap for more steps...

Step 6.4.1

Subtract $4$ from both sides of the equation.

$5 x = - 36 - 4$

Step 6.4.2

Subtract $4$ from $- 36$ .

$5 x = - 40$

Step 6.5

Divide each term in $5 x = - 40$ by $5$ and simplify.

Tap for more steps...

Step 6.5.1

Divide each term in $5 x = - 40$ by $5$ .

$\frac{5 x}{5} = \frac{- 40}{5}$

Step 6.5.2

Simplify the left side.

Tap for more steps...

Step 6.5.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 6.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 40}{5}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 40}{5}$

Step 6.5.3

Simplify the right side.

Tap for more steps...

Step 6.5.3.1

Divide $- 40$ by $5$ .

$x = - 8$