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

To simplify the expression \(\sqrt[3]{\frac{x \cdot x^{-\frac{1}{4}}}{\sqrt[4]{x^3}}}\), let's break it down step by step:

### Step 1: Simplify the numerator.
The numerator is \(x \cdot x^{-\frac{1}{4}}\).

Using the property of exponents \(a^m \cdot a^n = a^{m+n}\), we can combine the terms:

\[
x \cdot x^{-\frac{1}{4}} = x^{1 - \frac{1}{4}} = x^{\frac{3}{4}}
\]

So, the numerator simplifies to \(x^{\frac{3}{4}}\).

### Step 2: Simplify the denominator.
The denominator is \(\sqrt[4]{x^3}\), which can be rewritten using exponent notation:

\[
\sqrt[4]{x^3} = x^{\frac{3}{4}}
\]

### Step 3: Substitute the simplified expressions back into the original equation.
Now, substitute the simplified numerator and denominator back into the original expression:

\[
\sqrt[3]{\frac{x^{\frac{3}{4}}}{x^{\frac{3}{4}}}}
\]

### Step 4: Simplify the fraction inside the radical.
Since \(\frac{x^{\frac{3}{4}}}{x^{\frac{3}{4}}} = 1\), the expression simplifies to:

\[
\sqrt[3]{1}
\]

### Step 5: Simplify the cube root of 1.
The cube root of 1 is simply 1:

\[
\sqrt[3]{1} = 1
\]

So, the simplified expression is:

\[
1
\]

Would you like more details on any part of this process, or do you have any questions?

Here are five related questions you might be interested in:
1. How do you simplify expressions with fractional exponents?
2. What are the rules for multiplying and dividing expressions with exponents?
3. How can you simplify expressions with radicals of different orders?
4. How do you solve equations that involve cube roots?
5. What are some practical applications of simplifying expressions like this?

**Tip:** Always remember that when simplifying exponents, combining like terms and reducing fractions inside radicals are essential first steps.

Learn how to simplify cube roots of expressions involving fractional exponents with this detailed step-by-step guide. This problem involves exponents, radicals, and simplifying fractions inside radicals. Suitable for high school students.

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