Solve for x (1/3)^(x+2)=3^(x-1)

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

Apply the product rule to $\frac{1}{3}$ .

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

One to any power is one.

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

Move $3^{x + 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

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

$- (x + 2) = x - 1$

Solve for $x$ .

Simplify $- (x + 2)$ .

Rewrite.

$0 + 0 - (x + 2) = x - 1$

Simplify by adding zeros.

$- (x + 2) = x - 1$

Apply the distributive property.

$- x - 1 \cdot 2 = x - 1$

Multiply $- 1$ by $2$ .

$- x - 2 = x - 1$

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

Subtract $x$ from both sides of the equation.

$- x - 2 - x = - 1$

Subtract $x$ from $- x$ .

$- 2 x - 2 = - 1$

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

Add $2$ to both sides of the equation.

$- 2 x = - 1 + 2$

Add $- 1$ and $2$ .

$- 2 x = 1$

Divide each term in $- 2 x = 1$ by $- 2$ and simplify.

Divide each term in $- 2 x = 1$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.5$

Solve for x (1/3)^(x+2)=3^(x-1)

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