Solve for a 3^x-3^(x-2)=a*3^x

$3^{x} - 3^{x - 2} = a \cdot 3^{x}$

Step 1

Rewrite the equation as $a \cdot 3^{x} = 3^{x} - 3^{x - 2}$ .

$a \cdot 3^{x} = 3^{x} - 3^{x - 2}$

Step 2

Divide each term in $(a \cdot 3^{x}) = 3^{x} - 3^{x - 2}$ by $3^{x}$ and simplify.

Divide each term in $(a \cdot 3^{x}) = 3^{x} - 3^{x - 2}$ by $3^{x}$ .

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

Simplify the left side.

Cancel the common factor of $3^{x}$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

Split the fraction $\frac{3^{x} - 3^{x - 2}}{3^{x}}$ into two fractions.

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

Cancel the common factor of $3^{x}$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3^{x - 2}$ and $3^{x}$ .

Factor $3^{x}$ out of $- 3^{x - 2}$ .

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 3^{- 2}$ by $1$ .

$a = 1 - 3^{- 2}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$a = 1 - \frac{1}{3^{2}}$

Raise $3$ to the power of $2$ .

$a = 1 - \frac{1}{9}$

Write $1$ as a fraction with a common denominator.

$a = \frac{9}{9} - \frac{1}{9}$

Combine the numerators over the common denominator.

$a = \frac{9 - 1}{9}$

Subtract $1$ from $9$ .

$a = \frac{8}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$a = \frac{8}{9}$

Decimal Form:

$a = 0.\overline{8}$

Solve for a 3^x-3^(x-2)=a*3^x

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