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To simplify the given expression, let's go through it step-by-step.

The expression is:
\[
\frac{\left(x^{1/5} y^{4/5}\right)^{1/4}}{x^{1/10} y^{1/20}}
\]

### Step 1: Apply the Power Rule
Using the power rule \((a^m)^n = a^{m \cdot n}\), simplify the numerator:
\[
\left(x^{1/5} y^{4/5}\right)^{1/4} = x^{(1/5) \cdot (1/4)} y^{(4/5) \cdot (1/4)} = x^{1/20} y^{1/5}
\]

So now we have:
\[
\frac{x^{1/20} y^{1/5}}{x^{1/10} y^{1/20}}
\]

### Step 2: Simplify Each Variable Separately
Rewrite the expression as:
\[
\frac{x^{1/20}}{x^{1/10}} \cdot \frac{y^{1/5}}{y^{1/20}}
\]

#### Simplify the \(x\)-terms:
Using the rule \(\frac{a^m}{a^n} = a^{m-n}\),
\[
\frac{x^{1/20}}{x^{1/10}} = x^{1/20 - 1/10} = x^{1/20 - 2/20} = x^{-1/20} = \frac{1}{x^{1/20}}
\]

#### Simplify the \(y\)-terms:
Similarly,
\[
\frac{y^{1/5}}{y^{1/20}} = y^{1/5 - 1/20} = y^{4/20 - 1/20} = y^{3/20}
\]

### Step 3: Combine the Results
Now we have:
\[
\frac{1}{x^{1/20}} \cdot y^{3/20} = \frac{y^{3/20}}{x^{1/20}}
\]

### Final Answer
\[
\frac{y^{3/20}}{x^{1/20}}
\]

Would you like a breakdown of any step? 

### Related Questions
1. How do you simplify expressions involving fractional exponents?
2. What is the power rule in exponents, and how is it applied?
3. How do you handle negative exponents in simplification?
4. How can the expression \((a^m)^{n}\) be simplified further when \(m\) or \(n\) are fractions?
5. What is the difference between simplifying and evaluating an expression?

### Tip
When simplifying expressions with exponents, always consider whether any terms can cancel out by using the property \(\frac{a^m}{a^n} = a^{m-n}\) where possible.

Simplify: ((x^(1/5)y^(4/5))^(1/4)) / (x^(1/10)y^(1/20))

Learn how to simplify expressions with fractional exponents step-by-step. This problem demonstrates exponent laws, including the power rule and division of terms with fractional exponents, ideal for high school students in Grades 10-12.

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