Solve for x 3^(2x+1)*3^x=243

$3^{2 x + 1} \cdot 3^{x} = 243$

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

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

$3^{2 x + 1 + x} = 243$

Add $2 x$ and $x$ .

$3^{3 x + 1} = 243$

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

$3^{3 x + 1} = 3^{5}$

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

$3 x + 1 = 5$

Solve for $x$ .

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

Subtract $1$ from both sides of the equation.

$3 x = 5 - 1$

Subtract $1$ from $5$ .

$3 x = 4$

Divide each term in $3 x = 4$ by $3$ and simplify.

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Divide $x$ by $1$ .

$x = \frac{4}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}$

Decimal Form:

$x = 1.\overline{3}$

Mixed Number Form:

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