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

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

Step 1

Rewrite the equation as $9^{x} = 3^{\frac{2}{3}}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Divide each term in $2 x = \frac{2}{3}$ by $2$ and simplify.

Divide each term in $2 x = \frac{2}{3}$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}$