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To simplify the expression \(\sqrt[5]{x^6 y^7}\), we need to break down the exponents and apply the fifth root to each term individually.

### Step 1: Apply the fifth root to each variable

The expression is:

\[
\sqrt[5]{x^6 y^7}
\]

This can be rewritten as:

\[
\sqrt[5]{x^6} \cdot \sqrt[5]{y^7}
\]

### Step 2: Simplify each root

- For \(x^6\), the fifth root simplifies to:
  \[
  \sqrt[5]{x^6} = x^{6/5} = x^{1 + 1/5} = x \cdot x^{1/5}
  \]
  
- For \(y^7\), the fifth root simplifies to:
  \[
  \sqrt[5]{y^7} = y^{7/5} = y \cdot y^{2/5}
  \]

### Step 3: Combine the results

Now, multiply the simplified results together:

\[
x \cdot x^{1/5} \cdot y \cdot y^{2/5}
\]

This simplifies to:

\[
xy \cdot x^{1/5} \cdot y^{2/5} = xy \cdot x^{1/5}y^{2/5}
\]

### Final Answer:

The simplified expression is:

\[
xy \cdot \sqrt[5]{x} \cdot \sqrt[5]{y^2}
\]

Would you like further details or have any questions about this process?

Here are some related questions you might find interesting:
1. How do you simplify roots with higher exponents?
2. How would the simplification change if the exponents of \(x\) and \(y\) were different?
3. What is the process for rationalizing the denominator involving roots?
4. Can we simplify \(\sqrt[5]{x^6 y^7}\) using logarithms?
5. How does the simplification differ when dealing with cube roots instead of fifth roots?

**Tip:** When simplifying expressions involving roots, always break down the exponents first to see if any terms can be simplified further.

Learn how to simplify the expression \(\sqrt[5]{x^6 y^7}\) with step-by-step guidance. This problem involves algebraic concepts such as exponents and roots, applying the properties of exponents to simplify the expression. Suitable for students in Grades 10-12.

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