We are given the following expression:
[tex]\frac{\sqrt[4]{y^3}}{\sqrt[5]{y^3}}[/tex]
To convert to rational exponents we will use the following relationship:
[tex]\sqrt[y]{a^x}=a^{\frac{x}{y}}[/tex]
Applying the relationship we get;
[tex]\frac{\sqrt[4]{y^3}}{\sqrt[5]{y^3}}=\frac{y^{\frac{3}{4}}}{y^{\frac{3}{5}}}[/tex]
Now, we will use the following property of exponents on the denominator:
[tex]\frac{1}{a^x}=a^{-x}[/tex]
Therefore, we can bring the denominator up by inverting the sign of the exponents, like this:
[tex]\frac{y^{\frac{3}{4}}}{y^{\frac{3}{5}}}=y^{\frac{3}{4}}y^{-\frac{3}{5}}[/tex]
Now, we use the following property of exponents:
[tex]a^xa^y=a^{x+y}[/tex]
Applying the property we get:
[tex]y^{\frac{3}{4}}y^{-\frac{3}{5}}=y^{\frac{3}{4}-\frac{3}{5}}[/tex]
Adding the exponents we get:
[tex]y^{\frac{3}{4}-\frac{3}{5}}=y^{\frac{3}{20}}[/tex]
Since we can't simplify any further this is the final answer.
