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The given expression is:

\[
\frac{\sqrt[3]{x^{-\frac{1}{3}}}}{\sqrt[4]{y^{-\frac{1}{4}}}}
\]

### Step 1: Simplify the numerator and denominator.
We can express roots as exponents. So:
\[
\sqrt[3]{x^{-\frac{1}{3}}} = \left(x^{-\frac{1}{3}}\right)^{\frac{1}{3}} = x^{-\frac{1}{9}}
\]
And for the denominator:
\[
\sqrt[4]{y^{-\frac{1}{4}}} = \left(y^{-\frac{1}{4}}\right)^{\frac{1}{4}} = y^{-\frac{1}{16}}
\]

### Step 2: Combine the expression.
Now we have:
\[
\frac{x^{-\frac{1}{9}}}{y^{-\frac{1}{16}}}
\]

We can combine these into one expression by subtracting exponents:
\[
x^{-\frac{1}{9}} \cdot y^{\frac{1}{16}}
\]

### Final Expression:
\[
x^{-\frac{1}{9}} \cdot y^{\frac{1}{16}}
\]

This is the simplified form of the original expression.

Do you need more details, or any specific questions on this?

Here are some related questions to expand on this topic:
1. How do you simplify expressions involving fractional exponents?
2. What is the general rule for simplifying roots of variables with exponents?
3. Can the expression be rewritten as a single radical instead of using exponents?
4. How do you deal with negative exponents when simplifying expressions?
5. What is the relationship between radicals and exponents in mathematical operations?

**Tip:** When simplifying expressions with roots, remember that roots are just fractional exponents, making them easier to work with when applying the laws of exponents.

Simplify the expression: \frac{\sqrt[3]{x^{-\frac{1}{3}}}}{\sqrt[4]{y^{-\frac{1}{4}}}}.

Learn how to simplify an expression involving nested cube and fourth roots with fractional exponents: \frac{\sqrt[3]{x^{-1/3}}}{\sqrt[4]{y^{-1/4}}}. This step-by-step solution breaks down the exponentiation rules for students in Grades 10-12.

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