Algebra Examples

${(\frac{1}{243})}^{- \frac{x}{3}} = {(\frac{1}{9})}^{- \frac{x}{2} + 1}$

Step 1

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

${(\frac{1}{3})}^{5 (- \frac{x}{3})} = {(\frac{1}{3})}^{2 (- \frac{x}{2} + 1)}$

Step 2

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

$5 (- \frac{x}{3}) = 2 (- \frac{x}{2} + 1)$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Multiply $5 (- \frac{x}{3})$ .

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Step 3.1.1.1

Multiply $- 1$ by $5$ .

$- 5 \frac{x}{3} = 2 (- \frac{x}{2} + 1)$

Step 3.1.1.2

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

$\frac{- 5 x}{3} = 2 (- \frac{x}{2} + 1)$

Step 3.1.2

Move the negative in front of the fraction.

$- \frac{5 x}{3} = 2 (- \frac{x}{2} + 1)$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$- \frac{5 x}{3} = 2 (- \frac{x}{2}) + 2 \cdot 1$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$- \frac{5 x}{3} = 2 \frac{- x}{2} + 2 \cdot 1$

Step 3.2.2.2

Cancel the common factor.

$- \frac{5 x}{3} = 2 \frac{- x}{2} + 2 \cdot 1$

Step 3.2.2.3

Rewrite the expression.

$- \frac{5 x}{3} = - x + 2 \cdot 1$

Step 3.2.3

Multiply $2$ by $1$ .

$- \frac{5 x}{3} = - x + 2$

Step 3.3

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

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Step 3.3.1

Add $x$ to both sides of the equation.

$- \frac{5 x}{3} + x = 2$

Step 3.3.2

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

$- \frac{5 x}{3} + x \cdot \frac{3}{3} = 2$

Step 3.3.3

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

$- \frac{5 x}{3} + \frac{x \cdot 3}{3} = 2$

Step 3.3.4

Combine the numerators over the common denominator.

$\frac{- 5 x + x \cdot 3}{3} = 2$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Move $3$ to the left of $x$ .

$\frac{- 5 x + 3 \cdot x}{3} = 2$

Step 3.3.5.2

Add $- 5 x$ and $3 x$ .

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

Step 3.3.6

Move the negative in front of the fraction.

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

Step 3.4

Multiply both sides of the equation by $- \frac{3}{2}$ .

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

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

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

Step 3.5.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 3.5.1.1.1.4

Cancel the common factor.

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

Step 3.5.1.1.1.5

Rewrite the expression.

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

Step 3.5.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.1.1.2.1

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

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

Step 3.5.1.1.2.2

Cancel the common factor.

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

Step 3.5.1.1.2.3

Rewrite the expression.

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

Step 3.5.1.1.3

Multiply.

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Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.1.1.3.2

Multiply $x$ by $1$ .

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.5.2.1.2

Cancel the common factor.

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

Step 3.5.2.1.3

Rewrite the expression.

$x = - 3$