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?

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.