  \frac{\sqrt[n]{2*14^6}}{\sqrt[3]{2*14}\cdot \sqrt[3]{14^2}}

To simplify the expression 

\[
\frac{\sqrt[n]{2 \cdot 14^6}}{\sqrt[3]{2 \cdot 14} \cdot \sqrt[3]{14^2}},
\]

let's break it down step-by-step.

### Step 1: Simplify the Numerator

The numerator is \(\sqrt[n]{2 \cdot 14^6}\). This is an \(n\)-th root expression.

\[
\sqrt[n]{2 \cdot 14^6} = (2 \cdot 14^6)^{\frac{1}{n}}.
\]

### Step 2: Simplify the Denominator

The denominator is \(\sqrt[3]{2 \cdot 14} \cdot \sqrt[3]{14^2}\). These are cube root expressions, and they can be combined:

\[
\sqrt[3]{2 \cdot 14} \cdot \sqrt[3]{14^2} = \sqrt[3]{(2 \cdot 14) \cdot (14^2)}.
\]

Simplify the product inside the cube root:

\[
2 \cdot 14 \cdot 14^2 = 2 \cdot 14^3.
\]

So, the denominator simplifies to:

\[
\sqrt[3]{2 \cdot 14^3} = (2 \cdot 14^3)^{\frac{1}{3}}.
\]

### Step 3: Combine the Expressions

The full expression now is:

\[
\frac{(2 \cdot 14^6)^{\frac{1}{n}}}{(2 \cdot 14^3)^{\frac{1}{3}}}.
\]

### Step 4: Apply Exponent Laws

Write the expressions with exponents:

- Numerator: \((2 \cdot 14^6)^{\frac{1}{n}} = 2^{\frac{1}{n}} \cdot (14^6)^{\frac{1}{n}} = 2^{\frac{1}{n}} \cdot 14^{\frac{6}{n}}\).
- Denominator: \((2 \cdot 14^3)^{\frac{1}{3}} = 2^{\frac{1}{3}} \cdot (14^3)^{\frac{1}{3}} = 2^{\frac{1}{3}} \cdot 14^1 = 2^{\frac{1}{3}} \cdot 14\).

### Step 5: Simplify the Fraction

Now divide the expressions:

\[
\frac{2^{\frac{1}{n}} \cdot 14^{\frac{6}{n}}}{2^{\frac{1}{3}} \cdot 14}.
\]

Separate the terms:

\[
\frac{2^{\frac{1}{n}}}{2^{\frac{1}{3}}} \cdot \frac{14^{\frac{6}{n}}}{14^1}.
\]

Using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\), we get:

\[
2^{\frac{1}{n} - \frac{1}{3}} \cdot 14^{\frac{6}{n} - 1}.
\]

### Final Simplified Expression

\[
2^{\frac{1}{n} - \frac{1}{3}} \cdot 14^{\frac{6}{n} - 1}.
\]

This is the simplified form of the given expression.

### Summary

The expression simplifies to:

\[
2^{\frac{1}{n} - \frac{1}{3}} \cdot 14^{\frac{6}{n} - 1}.
\]

Would you like further details or have any questions?

### Related Questions:

1. How do you simplify expressions involving higher roots?
2. What are the exponent rules for dividing powers with the same base?
3. How does the simplification change if \(n = 3\)?
4. What happens if you have negative exponents in a root?
5. How would you express this result as a decimal or fraction for specific \(n\) values?

#### Tip:

When simplifying expressions involving roots and powers, always look to combine like terms and use exponent rules to reduce the expression to its simplest form.

Learn how to simplify the expression \(\frac{\sqrt[n]{2 \cdot 14^6}}{\sqrt[3]{2 \cdot 14} \cdot \sqrt[3]{14^2}}\) step-by-step with detailed explanations and examples. This problem involves roots, radicals, and exponent rules, suitable for advanced high school students and those preparing for mathematical competitions.

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