Algebra Examples

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

Step 1

Multiply $3^{- 2 x + 1}$ by $3^{- 2 x - 3}$ by adding the exponents.

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Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Subtract $2 x$ from $- 2 x$ .

$3^{- 4 x + 1 - 3} = 3^{- x}$

Subtract $3$ from $1$ .

$3^{- 4 x - 2} = 3^{- x}$

Step 2

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

$- 4 x - 2 = - x$

Step 3

Solve for $x$ .

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

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Add $x$ to both sides of the equation.

$- 4 x - 2 + x = 0$

Add $- 4 x$ and $x$ .

$- 3 x - 2 = 0$

Add $2$ to both sides of the equation.

$- 3 x = 2$

Divide each term in $- 3 x = 2$ by $- 3$ and simplify.

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Divide each term in $- 3 x = 2$ by $- 3$ .

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

Simplify the left side.

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Cancel the common factor of $- 3$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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Move the negative in front of the fraction.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.\overline{6}$