Algebra Examples

$32^{\frac{x}{3}} = 8^{x - 12}$

Step 1

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

$2^{5 \frac{x}{3}} = 2^{3 (x - 12)}$

Step 2

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

$5 \frac{x}{3} = 3 (x - 12)$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify $5 (\frac{x}{3})$ .

Tap for more steps...

Step 3.1.1

Rewrite.

$0 + 0 + 5 (\frac{x}{3}) = 3 (x - 12)$

Step 3.1.2

Simplify by adding zeros.

$5 (\frac{x}{3}) = 3 (x - 12)$

Step 3.1.3

Combine $5$ and $\frac{x}{3}$ .

$\frac{5 x}{3} = 3 (x - 12)$

Step 3.2

Simplify $3 (x - 12)$ .

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$\frac{5 x}{3} = 3 x + 3 \cdot - 12$

Step 3.2.2

Multiply $3$ by $- 12$ .

$\frac{5 x}{3} = 3 x - 36$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$\frac{5 x}{3} - 3 x = - 36$

Step 3.3.2

To write $- 3 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 x}{3} - 3 x \cdot \frac{3}{3} = - 36$

Step 3.3.3

Combine $- 3 x$ and $\frac{3}{3}$ .

$\frac{5 x}{3} + \frac{- 3 x \cdot 3}{3} = - 36$

Step 3.3.4

Combine the numerators over the common denominator.

$\frac{5 x - 3 x \cdot 3}{3} = - 36$

Step 3.3.5

Simplify the numerator.

Tap for more steps...

Step 3.3.5.1

Factor $x$ out of $5 x - 3 x \cdot 3$ .

Tap for more steps...

Step 3.3.5.1.1

Factor $x$ out of $5 x$ .

$\frac{x \cdot 5 - 3 x \cdot 3}{3} = - 36$

Step 3.3.5.1.2

Factor $x$ out of $- 3 x \cdot 3$ .

$\frac{x \cdot 5 + x (- 3 \cdot 3)}{3} = - 36$

Step 3.3.5.1.3

Factor $x$ out of $x \cdot 5 + x (- 3 \cdot 3)$ .

$\frac{x (5 - 3 \cdot 3)}{3} = - 36$

Step 3.3.5.2

Multiply $- 3$ by $3$ .

$\frac{x (5 - 9)}{3} = - 36$

Step 3.3.5.3

Subtract $9$ from $5$ .

$\frac{x \cdot - 4}{3} = - 36$

Step 3.3.6

Move $- 4$ to the left of $x$ .

$\frac{- 4 \cdot x}{3} = - 36$

Step 3.3.7

Move the negative in front of the fraction.

$- \frac{4 x}{3} = - 36$

Step 3.4

Multiply both sides of the equation by $- \frac{3}{4}$ .

$- \frac{3}{4} (- \frac{4 x}{3}) = - \frac{3}{4} \cdot - 36$

Step 3.5

Simplify both sides of the equation.

Tap for more steps...

Step 3.5.1

Simplify the left side.

Tap for more steps...

Step 3.5.1.1

Simplify $- \frac{3}{4} (- \frac{4 x}{3})$ .

Tap for more steps...

Step 3.5.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.1.1.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} (- \frac{4 x}{3}) = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{4 x}{3}$ into the numerator.

$\frac{- 3}{4} \cdot \frac{- 4 x}{3} = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{4} \cdot \frac{- 4 x}{3} = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{4} \cdot \frac{- 4 x}{3} = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 x) = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.5.1.1.2.1

Factor $4$ out of $- 4 x$ .

$\frac{- 1}{4} (4 (- x)) = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- x)) = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.2.3

Rewrite the expression.

$- - x = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.3

Multiply.

Tap for more steps...

Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{4} \cdot - 36$

Step 3.5.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{4} \cdot - 36$

Step 3.5.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.1

Simplify $- \frac{3}{4} \cdot - 36$ .

Tap for more steps...

Step 3.5.2.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.5.2.1.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$x = \frac{- 3}{4} \cdot - 36$

Step 3.5.2.1.1.2

Factor $4$ out of $- 36$ .

$x = \frac{- 3}{4} \cdot (4 (- 9))$

Step 3.5.2.1.1.3

Cancel the common factor.

$x = \frac{- 3}{4} \cdot (4 \cdot - 9)$

Step 3.5.2.1.1.4

Rewrite the expression.

$x = - 3 \cdot - 9$

Step 3.5.2.1.2

Multiply $- 3$ by $- 9$ .

$x = 27$