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Let's go through each problem one by one to simplify the expressions:

### a) \(\sqrt[3]{x \cdot x^{1/6}}\)
We can apply the property of exponents: \(\sqrt[n]{a^m} = a^{m/n}\).

1. Combine the exponents inside the root: \(x \cdot x^{1/6} = x^{1 + 1/6} = x^{6/6 + 1/6} = x^{7/6}\).
2. Apply the cube root: \(\sqrt[3]{x^{7/6}} = x^{7/18}\).

### b) \(\sqrt[3]{(\sqrt{n})^4}\)
1. First simplify the square root: \(\sqrt{n} = n^{1/2}\).
2. Raise \(n^{1/2}\) to the power of 4: \((n^{1/2})^4 = n^{4/2} = n^2\).
3. Now apply the cube root: \(\sqrt[3]{n^2} = n^{2/3}\).

### c) \(3 \sqrt{75} \cdot 4 \sqrt[4]{48}\)
1. Simplify \(\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}\).
2. Simplify \(\sqrt[4]{48} = \sqrt[4]{16 \cdot 3} = 2\sqrt[4]{3}\).
3. Now multiply: 
   \[
   3 \cdot 5\sqrt{3} = 15\sqrt{3},
   \]
   and 
   \[
   4 \cdot 2\sqrt[4]{3} = 8\sqrt[4]{3}.
   \]
4. The final expression is \(15\sqrt{3} \cdot 8\sqrt[4]{3} = 120\sqrt[4]{3^5} = 120\cdot 3^{5/4}\).

### d) \(\sqrt{\frac{12}{10}} \cdot \sqrt{\frac{27}{40}}\)
1. Simplify the fractions: \(\frac{12}{10} = \frac{6}{5}\) and \(\frac{27}{40}\) cannot be simplified.
2. Multiply inside the square roots: 
   \[
   \sqrt{\frac{6}{5}} \cdot \sqrt{\frac{27}{40}} = \sqrt{\frac{6 \cdot 27}{5 \cdot 40}} = \sqrt{\frac{162}{200}} = \sqrt{\frac{81}{100}} = \frac{9}{10}.
   \]

### e) \(\left( \sqrt{\sqrt{2}} \right)^4\)
1. First, simplify \(\sqrt{\sqrt{2}} = 2^{1/4}\).
2. Raise \(2^{1/4}\) to the power of 4: \((2^{1/4})^4 = 2^{4/4} = 2\).

### f) \(\frac{\sqrt{x^3 y^2}}{z^6} \cdot \sqrt[3]{\frac{x^2 y^3}{z^4}}\)
1. Simplify the first term: \(\sqrt{x^3 y^2} = x^{3/2} y\).
2. Simplify the second term: \(\sqrt[3]{\frac{x^2 y^3}{z^4}} = \frac{x^{2/3} y z^{4/3}}\).
3. Combine the terms: 
   \[
   \frac{x^{3/2} y}{z^6} \cdot \frac{x^{2/3} y}{z^{4/3}} = x^{3/2 + 2/3} \cdot y^2 \cdot z^{-6 - 4/3}.
   \]
4. Add the exponents: \(x^{13/6} \cdot y^2 \cdot z^{-22/3}\).

### g) \(\sqrt[3]{x^2} \cdot \sqrt[3]{\frac{1}{x^4}} \cdot x^5\)
1. Simplify \(\sqrt[3]{x^2} = x^{2/3}\), and \(\sqrt[3]{\frac{1}{x^4}} = x^{-4/3}\).
2. Multiply the terms: 
   \[
   x^{2/3} \cdot x^{-4/3} \cdot x^5 = x^{(2/3 - 4/3 + 5)} = x^{13/3}.
   \]

### h) \((\sqrt{2} \cdot \sqrt{2})^3\)
1. Simplify inside the parentheses: \(\sqrt{2} \cdot \sqrt{2} = 2\).
2. Now raise to the power of 3: \(2^3 = 8\).

### i) \(\frac{5}{3} \cdot \sqrt[3]{\frac{1}{a^4}} \cdot a^{1/3}\)
1. Simplify \(\sqrt[3]{\frac{1}{a^4}} = a^{-4/3}\).
2. Now multiply the terms: 
   \[
   \frac{5}{3} \cdot a^{-4/3 + 1/3} = \frac{5}{3} \cdot a^{-3/3} = \frac{5}{3} \cdot a^{-1}.
   \]
3. Final expression: \(\frac{5}{3a}\).

### j) \(\frac{a^{-1/2}}{\sqrt{a}} \cdot \frac{a^{3/4}}{\sqrt[4]{a^2}}\)
1. Simplify each term: \(\sqrt{a} = a^{1/2}\) and \(\sqrt[4]{a^2} = a^{1/2}\).
2. Now rewrite:
   \[
   \frac{a^{-1/2}}{a^{1/2}} \cdot \frac{a^{3/4}}{a^{1/2}} = a^{-1} \cdot a^{1/4} = a^{-3/4}.
   \]

### k) \(\frac{a^4 \cdot a^{-1/3} \cdot \sqrt[3]{a}}{3 \cdot \sqrt{a}}\)
1. First, simplify \(\sqrt[3]{a} = a^{1/3}\) and \(\sqrt{a} = a^{1/2}\).
2. Now combine the terms in the numerator: 
   \[
   a^4 \cdot a^{-1/3} \cdot a^{1/3} = a^4.
   \]
3. Combine the whole expression: 
   \[
   \frac{a^4}{3a^{1/2}} = \frac{a^{4 - 1/2}}{3} = \frac{a^{7/2}}{3}.
   \]

Would you like more details on any of these steps or have any questions? 

Here are some follow-up questions based on these problems:
1. What is the general rule for combining exponents when multiplying like bases?
2. How do you convert a radical expression into exponential form?
3. What steps are involved in simplifying nested radicals?
4. Can you describe the process of rationalizing denominators?
5. How can you apply cube roots and higher roots in simplifications?

**Tip**: When simplifying radicals, always try converting them into fractional exponents—it often makes the calculations easier to handle.

Efectúa y simplifica si es posible:
a) ³√x⋅x^(1/6)
b) ³√(√n)^4
c) 3√75 ⋅ 4√⁴48
d) √(12/10) ⋅ √(27/40)
e) (√√2)^4
f) √(x³y²)/z⁶ ⋅ ³√(x²y³/z⁴)
g) ³√x² ⋅ ³√(1/x⁴) ⋅ x⁵
h) (√2√2)³
i) 5/3 ⋅ ³√(1/a⁴) ⋅ a^(1/3)
j) a^(-1/2)/√a ⋅ a^(3/4)/√⁴(a²)
k) (a⁴ ⋅ a^(-1/3) ⋅ ³√a)/(3√a)

This tutorial simplifies complex expressions involving radicals, exponents, and cube roots. Problems include operations such as cube roots of products, simplifying square roots of fractions, and applying laws of exponents. Suitable for high school students in Grades 10-12.

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