Let's simplify the expression:

\[
\frac{k^2m + k^2m^3}{m^3k}
\]

### Step 1: Factor the numerator
First, factor out the common factor from the terms in the numerator, \( k^2 \):

\[
k^2m + k^2m^3 = k^2(m + m^3)
\]

### Step 2: Rewrite the expression
Now the expression becomes:

\[
\frac{k^2(m + m^3)}{m^3k}
\]

### Step 3: Simplify the fraction
Cancel out the common \( k \) factor:

\[
\frac{k^2}{k} = k
\]

This gives us:

\[
\frac{k(m + m^3)}{m^3}
\]

### Step 4: Simplify further
Now simplify each term of \( \frac{m + m^3}{m^3} \):

\[
\frac{m}{m^3} = \frac{1}{m^2}, \quad \frac{m^3}{m^3} = 1
\]

Thus, \( \frac{m + m^3}{m^3} = \frac{1}{m^2} + 1 \).

### Step 5: Final expression
The simplified expression is:

\[
\frac{k(m + m^3)}{m^3} = k\left( \frac{1}{m^2} + 1 \right)
\]

Or:

\[
k\left( 1 + \frac{1}{m^2} \right)
\]

This is the final simplified form.

Would you like further details on any step?

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Here are 5 related questions:
1. How do you simplify fractions involving variables?
2. What are common factoring techniques for polynomials?
3. How do you cancel terms in algebraic fractions?
4. What is the process for factoring out the greatest common factor (GCF)?
5. How does the power of variables affect simplification?

**Tip:** Always look for common factors in expressions before simplifying further.

Learn how to simplify the expression k^2m + k^2m^3 divided by m^3k with a step-by-step breakdown of algebraic techniques like factoring and fraction simplification. Suitable for students in Grades 8-10.

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