Simplify. \[ \frac{y^{\frac{5}{9}}}{y^{\frac{2}{3}}} \] Write your answer using only a positive exponent. Assume that the variable represents a positive real number.

To simplify the expression \[ \frac{y^{\frac{5}{9}}}{y^{\frac{2}{3}}} \] we can use the property of exponents that states \[ \frac{a^m}{a^n} = a^{m-n}. \] In this case, we have: - \( m = \frac{5}{9} \) - \( n = \frac{2}{3} \) First, we need to express \( \frac{2}{3} \) with a denominator of 9 to perform the subtraction. We can convert \( \frac{2}{3} \) as follows: \[ \frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}. \] Now we can substitute this back into our expression: \[ \frac{y^{\frac{5}{9}}}{y^{\frac{2}{3}}} = \frac{y^{\frac{5}{9}}}{y^{\frac{6}{9}}}. \] Now we apply the exponent rule: \[ y^{\frac{5}{9} - \frac{6}{9}} = y^{\frac{5 - 6}{9}} = y^{-\frac{1}{9}}. \] Since we want to express the answer using only a positive exponent, we can rewrite \( y^{-\frac{1}{9}} \) as: \[ \frac{1}{y^{\frac{1}{9}}}. \] Thus, the simplified expression is \[ \boxed{\frac{1}{y^{\frac{1}{9}}}}. \]