Algebra Examples

${(\frac{1}{8})}^{\frac{x}{2} - 1} = {(\frac{1}{4})}^{\frac{x}{3}}$

Step 1

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

${(\frac{1}{2})}^{3 (\frac{x}{2} - 1)} = {(\frac{1}{2})}^{2 \frac{x}{3}}$

Step 2

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

$3 (\frac{x}{2} - 1) = 2 \frac{x}{3}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify $3 (\frac{x}{2} - 1)$ .

Tap for more steps...

Step 3.1.1

Rewrite.

$0 + 0 + 3 (\frac{x}{2} - 1) = 2 (\frac{x}{3})$

Step 3.1.2

Simplify by adding zeros.

$3 (\frac{x}{2} - 1) = 2 (\frac{x}{3})$

Step 3.1.3

Apply the distributive property.

$3 \frac{x}{2} + 3 \cdot - 1 = 2 (\frac{x}{3})$

Step 3.1.4

Combine $3$ and $\frac{x}{2}$ .

$\frac{3 x}{2} + 3 \cdot - 1 = 2 (\frac{x}{3})$

Step 3.1.5

Multiply $3$ by $- 1$ .

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

Step 3.2

Combine $2$ and $\frac{x}{3}$ .

$\frac{3 x}{2} - 3 = \frac{2 x}{3}$

Step 3.3

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

Tap for more steps...

Step 3.3.1

Subtract $\frac{2 x}{3}$ from both sides of the equation.

$\frac{3 x}{2} - 3 - \frac{2 x}{3} = 0$

Step 3.3.2

To write $\frac{3 x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x}{2} \cdot \frac{3}{3} - \frac{2 x}{3} - 3 = 0$

Step 3.3.3

To write $- \frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{2} \cdot \frac{3}{3} - \frac{2 x}{3} \cdot \frac{2}{2} - 3 = 0$

Step 3.3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.3.4.1

Multiply $\frac{3 x}{2}$ by $\frac{3}{3}$ .

$\frac{3 x \cdot 3}{2 \cdot 3} - \frac{2 x}{3} \cdot \frac{2}{2} - 3 = 0$

Step 3.3.4.2

Multiply $2$ by $3$ .

$\frac{3 x \cdot 3}{6} - \frac{2 x}{3} \cdot \frac{2}{2} - 3 = 0$

Step 3.3.4.3

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$\frac{3 x \cdot 3}{6} - \frac{2 x \cdot 2}{3 \cdot 2} - 3 = 0$

Step 3.3.4.4

Multiply $3$ by $2$ .

$\frac{3 x \cdot 3}{6} - \frac{2 x \cdot 2}{6} - 3 = 0$

Step 3.3.5

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 - 2 x \cdot 2}{6} - 3 = 0$

Step 3.3.6

Simplify each term.

Tap for more steps...

Step 3.3.6.1

Simplify the numerator.

Tap for more steps...

Step 3.3.6.1.1

Factor $x$ out of $3 x \cdot 3 - 2 x \cdot 2$ .

Tap for more steps...

Step 3.3.6.1.1.1

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

$\frac{x (3 \cdot 3) - 2 x \cdot 2}{6} - 3 = 0$

Step 3.3.6.1.1.2

Factor $x$ out of $- 2 x \cdot 2$ .

$\frac{x (3 \cdot 3) + x (- 2 \cdot 2)}{6} - 3 = 0$

Step 3.3.6.1.1.3

Factor $x$ out of $x (3 \cdot 3) + x (- 2 \cdot 2)$ .

$\frac{x (3 \cdot 3 - 2 \cdot 2)}{6} - 3 = 0$

Step 3.3.6.1.2

Multiply $3$ by $3$ .

$\frac{x (9 - 2 \cdot 2)}{6} - 3 = 0$

Step 3.3.6.1.3

Multiply $- 2$ by $2$ .

$\frac{x (9 - 4)}{6} - 3 = 0$

Step 3.3.6.1.4

Subtract $4$ from $9$ .

$\frac{x \cdot 5}{6} - 3 = 0$

Step 3.3.6.2

Move $5$ to the left of $x$ .

$\frac{5 x}{6} - 3 = 0$

Step 3.4

Add $3$ to both sides of the equation.

$\frac{5 x}{6} = 3$

Step 3.5

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 3$

Step 3.6

Simplify both sides of the equation.

Tap for more steps...

Step 3.6.1

Simplify the left side.

Tap for more steps...

Step 3.6.1.1

Simplify $\frac{6}{5} \cdot \frac{5 x}{6}$ .

Tap for more steps...

Step 3.6.1.1.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.6.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 3$

Step 3.6.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 3$

Step 3.6.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.6.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} \cdot 3$

Step 3.6.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 3$

Step 3.6.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5} \cdot 3$

Step 3.6.2

Simplify the right side.

Tap for more steps...

Step 3.6.2.1

Multiply $\frac{6}{5} \cdot 3$ .

Tap for more steps...

Step 3.6.2.1.1

Combine $\frac{6}{5}$ and $3$ .

$x = \frac{6 \cdot 3}{5}$

Step 3.6.2.1.2

Multiply $6$ by $3$ .

$x = \frac{18}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{18}{5}$

Decimal Form:

$x = 3.6$

Mixed Number Form:

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