Solve for x (3^(x/2))/9=3(3^(2x))

$\frac{3^{\frac{x}{2}}}{9} = 3 (3^{2 x})$

Raise $9$ to the power of $1$ .

$\frac{3^{\frac{x}{2}}}{9^{1}} = 3 \cdot 3^{2 x}$

Move $9^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{\frac{x}{2}} \cdot 9^{- 1} = 3 \cdot 3^{2 x}$

Rewrite $9$ as $3^{2}$ .

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

Multiply the exponents in ${(3^{2})}^{- 1}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $- 1$ .

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

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

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

Multiply $3$ by $3^{2 x}$ .

Raise $3$ to the power of $1$ .

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

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

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

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

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

Solve for $x$ .

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

Subtract $2 x$ from both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

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

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

Factor $x$ out of $x \cdot 1 + x (- 2 \cdot 2)$ .

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

Multiply $- 2$ by $2$ .

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

Subtract $4$ from $1$ .

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

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

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

Move the negative in front of the fraction.

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

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

Add $2$ to both sides of the equation.

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

Add $1$ and $2$ .

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

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $2$ .

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

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

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

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$x = - 2$

Solve for x (3^(x/2))/9=3(3^(2x))

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