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To solve the problem of reducing the given roots to a common index, we'll first analyze the expression:

\[
\sqrt{14} \cdot \sqrt[3]{52}
\]

### Step 1: Convert roots to exponential form

The square root \(\sqrt{14}\) can be expressed as \(14^{1/2}\) and the cube root \(\sqrt[3]{52}\) as \(52^{1/3}\).

So, the expression becomes:
\[
14^{1/2} \cdot 52^{1/3}
\]

### Step 2: Find the Least Common Multiple (LCM) of the indices

The indices here are 2 and 3. The LCM of 2 and 3 is 6.

### Step 3: Rewrite each term with a common index of 6

We want to express each root with an exponent that has a denominator of 6:

\[
14^{1/2} = 14^{3/6} \quad \text{and} \quad 52^{1/3} = 52^{2/6}
\]

### Step 4: Combine the expressions

Now that the indices are the same, we can combine the terms:

\[
14^{3/6} \cdot 52^{2/6} = (14^3 \cdot 52^2)^{1/6}
\]

### Step 5: Simplify inside the parentheses

Calculate \(14^3\) and \(52^2\):

- \(14^3 = 14 \times 14 \times 14 = 2744\)
- \(52^2 = 52 \times 52 = 2704\)

So the expression becomes:

\[
(2744 \cdot 2704)^{1/6}
\]

Finally, multiply the numbers inside:

\[
2744 \times 2704 = 7410176
\]

Thus, the final expression in common index form is:

\[
\sqrt[6]{7410176}
\]

Would you like to ask anything else or need further details on any step?

### Related Questions:
1. How do you find the least common multiple (LCM) of two numbers?
2. Can you explain the process of converting roots to exponential form?
3. What happens when the LCM of indices is the same?
4. How do you multiply large numbers efficiently?
5. Why do we need to convert to a common index when dealing with different roots?
6. What are the steps to simplify roots with different indices?
7. Can we further simplify \(\sqrt[6]{7410176}\)?
8. How would the approach change if the indices were not integers?

### Tip:
When working with roots, always check if the expressions inside the roots can be simplified before or after converting to a common index to make calculations easier.

Learn how to simplify roots with different indices using a step-by-step guide. This problem involves converting roots to exponential form, finding the LCM of indices, and simplifying expressions. Suitable for high school students and anyone interested in advanced mathematical techniques.

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