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To simplify this expression, we can first rewrite each component using fractional exponents. Here’s how it works:

### Given Expression
\[
\frac{\sqrt[3]{a} \cdot \sqrt[3]{a^2}}{\sqrt[4]{a^3} \cdot \sqrt[4]{a}}
\]

### Step 1: Convert to Exponents
Using the rule that \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\), we can rewrite each term:

1. \(\sqrt[3]{a} = a^{\frac{1}{3}}\)
2. \(\sqrt[3]{a^2} = a^{\frac{2}{3}}\)
3. \(\sqrt[4]{a^3} = a^{\frac{3}{4}}\)
4. \(\sqrt[4]{a} = a^{\frac{1}{4}}\)

### Step 2: Substitute and Simplify
Substituting these values back into the expression, we get:

\[
\frac{a^{\frac{1}{3}} \cdot a^{\frac{2}{3}}}{a^{\frac{3}{4}} \cdot a^{\frac{1}{4}}}
\]

### Step 3: Combine Exponents
Using the property \(a^m \cdot a^n = a^{m+n}\), we can simplify the numerator and denominator separately:

1. **Numerator:** \(a^{\frac{1}{3} + \frac{2}{3}} = a^{\frac{3}{3}} = a^1 = a\)
2. **Denominator:** \(a^{\frac{3}{4} + \frac{1}{4}} = a^{\frac{4}{4}} = a^1 = a\)

So the expression simplifies to:

\[
\frac{a}{a} = 1
\]

### Final Answer
\[
1
\]

Would you like further details or have any questions?

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Here are some related questions for practice:

1. How would you simplify \(\frac{\sqrt[3]{b^2} \cdot \sqrt[3]{b^3}}{\sqrt[4]{b^5} \cdot \sqrt[4]{b}}\)?
2. What is the value of \(\frac{\sqrt[5]{x^4} \cdot \sqrt[5]{x}}{\sqrt[6]{x^5}}\)?
3. How do you handle fractional exponents in expressions like \(\frac{\sqrt[4]{y^3}}{\sqrt[6]{y}}\)?
4. How can you simplify \(\sqrt[4]{a^8} \cdot \sqrt[3]{a^9}\)?
5. What is the general rule for simplifying expressions involving radicals and powers?

**Tip:** When working with radicals, converting to fractional exponents can often make simplification easier, especially when multiplying or dividing terms.

Simplify the expression \( \frac{\sqrt[3]{a} \cdot \sqrt[3]{a^2}}{\sqrt[4]{a^3} \cdot \sqrt[4]{a}} \).

This page provides a step-by-step solution for simplifying the radical expression \( \frac{\sqrt[3]{a} \cdot \sqrt[3]{a^2}}{\sqrt[4]{a^3} \cdot \sqrt[4]{a}} \), covering key concepts such as fractional exponents and exponent rules. Suitable for high school students in Grades 10-12.

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