Solve for y (3^3)/(3^y)=1/9

$\frac{3^{3}}{3^{y}} = \frac{1}{9}$

Cancel the common factor of $3^{3}$ and $3^{y}$ .

Factor $3^{y}$ out of $3^{3}$ .

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{3^{3 - y}}{1} = \frac{1}{9}$

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

$3^{3 - y} = \frac{1}{9}$

Solve for $y$ .

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

$3^{3 - y} = \frac{1}{9^{1}}$

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

$3^{3 - y} = 9^{- 1}$

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

$3^{3 - y} = 3^{- 2}$

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

$3 - y = - 2$

Solve for $y$ .

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

Subtract $3$ from both sides of the equation.

$- y = - 2 - 3$

Subtract $3$ from $- 2$ .

$- y = - 5$

Divide each term in $- y = - 5$ by $- 1$ and simplify.

Divide each term in $- y = - 5$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 5}{- 1}$

Divide $y$ by $1$ .

$y = \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$y = 5$