To simplify the given expression:
[tex]\[
\frac{\left(y^3\right)^{\frac{-2}{3}}\left(y^2\right)^{\frac{1}{3}}}{\left(y^{\frac{1}{3}}\right)^4}
\][/tex]
first consider each term separately and simplify the exponents:
1. [tex]\(\left(y^3\right)^{\frac{-2}{3}}\)[/tex]:
[tex]\[
y^{3 \cdot \frac{-2}{3}} = y^{-2}
\][/tex]
2. [tex]\(\left(y^2\right)^{\frac{1}{3}}\)[/tex]:
[tex]\[
y^{2 \cdot \frac{1}{3}} = y^{\frac{2}{3}}
\][/tex]
3. [tex]\(\left(y^{\frac{1}{3}}\right)^4\)[/tex]:
[tex]\[
y^{\frac{1}{3} \cdot 4} = y^{\frac{4}{3}}
\][/tex]
Rewrite the expression with these simplified components:
[tex]\[
\frac{y^{-2} \cdot y^{\frac{2}{3}}}{y^{\frac{4}{3}}}
\][/tex]
Combine the exponents in the numerator:
[tex]\[
y^{-2 + \frac{2}{3}}
\][/tex]
To combine these exponents, find a common denominator:
[tex]\[
-2 + \frac{2}{3} = -\frac{6}{3} + \frac{2}{3} = -\frac{4}{3}
\][/tex]
So the simplified numerator is:
[tex]\[
y^{-\frac{4}{3}}
\][/tex]
Now the expression is:
[tex]\[
\frac{y^{-\frac{4}{3}}}{y^{\frac{4}{3}}}
\][/tex]
Subtract the exponents since it's a division:
[tex]\[
y^{-\frac{4}{3} - \frac{4}{3}} = y^{-\frac{8}{3}}
\][/tex]
Write using positive exponents:
[tex]\[
\frac{1}{y^{\frac{8}{3}}}
\][/tex]
Thus, the given expression simplifies to:
[tex]\[
\boxed{\frac{1}{y^{\frac{8}{3}}}}
\][/tex]
