\frac{\left(x^kx^{m-k}\right)^n}{\left(x^{2n}\right)^m}

We are tasked with simplifying the expression:

\[
\frac{\left(x^k x^{m-k}\right)^n}{\left(x^{2n}\right)^m}
\]

### Step 1: Simplify the numerator
The numerator is \(\left(x^k x^{m-k}\right)^n\).

Using the property of exponents that states \(x^a x^b = x^{a+b}\), we combine the powers inside the parentheses:

\[
x^k x^{m-k} = x^{k + (m - k)} = x^m
\]

Now, raise this result to the power of \(n\):

\[
\left(x^m\right)^n = x^{mn}
\]

So the numerator simplifies to \(x^{mn}\).

### Step 2: Simplify the denominator
The denominator is \(\left(x^{2n}\right)^m\).

Using the property of exponents \(\left(x^a\right)^b = x^{ab}\), we raise \(x^{2n}\) to the power of \(m\):

\[
\left(x^{2n}\right)^m = x^{2nm}
\]

### Step 3: Combine the results
We now have:

\[
\frac{x^{mn}}{x^{2nm}}
\]

Using the quotient rule of exponents \(\frac{x^a}{x^b} = x^{a-b}\), we subtract the exponents:

\[
x^{mn - 2nm} = x^{-mn}
\]

Thus, the simplified expression is:

\[
x^{-mn}
\]

Alternatively, this can be written as:

\[
\frac{1}{x^{mn}}
\]

### Final Answer:

\[
\boxed{\frac{1}{x^{mn}}}
\]

---

Would you like any additional details or explanations? Here are five related questions you might explore:

1. What are the laws of exponents used in this problem?
2. How do you simplify expressions with multiple exponents?
3. What is the meaning of a negative exponent?
4. How does raising a product of terms to a power work?
5. How does the quotient rule for exponents function?

**Tip**: Always simplify the expression inside the parentheses before applying the exponent!

This detailed solution simplifies the expression \frac{\left(x^kx^{m-k}\right)^n}{\left(x^{2n}\right)^m} using the laws of exponents, focusing on how to apply exponent rules and quotient rule. Ideal for students in Grades 9-12 learning algebraic exponent manipulation.

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