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To simplify the expression by introducing factors under the radicals, let's work through the given expression step-by-step. The expression is:

\[
0.1m^{\frac{4}{3}} \sqrt[3]{10m^2}
\]

### Step-by-Step Simplification:

1. **Rewrite the expression using exponents:**

   The cube root can be written as an exponent:
   
   \[
   0.1m^{\frac{4}{3}} (10m^2)^{\frac{1}{3}}
   \]

2. **Distribute the exponent inside the cube root:**

   Applying the exponent \(\frac{1}{3}\) to both \(10\) and \(m^2\):

   \[
   0.1m^{\frac{4}{3}} \cdot 10^{\frac{1}{3}} \cdot (m^2)^{\frac{1}{3}}
   \]

3. **Simplify the exponents of \(m\):**

   \((m^2)^{\frac{1}{3}} = m^{\frac{2}{3}}\)

   So the expression becomes:

   \[
   0.1 \cdot m^{\frac{4}{3}} \cdot 10^{\frac{1}{3}} \cdot m^{\frac{2}{3}}
   \]

4. **Combine like terms for \(m\):**

   Combine the exponents of \(m\):

   \[
   m^{\frac{4}{3} + \frac{2}{3}} = m^{\frac{6}{3}} = m^2
   \]

   So now the expression is:

   \[
   0.1 \cdot 10^{\frac{1}{3}} \cdot m^2
   \]

5. **Simplify the constants:**

   \[
   0.1 = 10^{-1}
   \]
   
   So the expression now becomes:

   \[
   10^{-1} \cdot 10^{\frac{1}{3}} \cdot m^2
   \]

6. **Combine the powers of \(10\):**

   Add the exponents:

   \[
   10^{-1 + \frac{1}{3}} \cdot m^2 = 10^{-\frac{2}{3}} \cdot m^2
   \]

   Thus, the simplified form with all factors introduced under the radicals is:

   \[
   10^{-\frac{2}{3}} m^2
   \]

   If you want this expressed as a radical:

   \[
   \frac{m^2}{\sqrt[3]{10^2}} = \frac{m^2}{\sqrt[3]{100}}
   \]

### Final Answer:

The simplified expression with factors under the radicals is:

\[
\frac{m^2}{\sqrt[3]{100}}
\]

Would you like a more detailed explanation, or do you have any questions about this process? Here are some related topics you might find interesting:

1. How to simplify expressions with higher roots (like fourth or fifth roots).
2. Techniques for simplifying expressions involving multiple radicals.
3. Understanding the rules of exponents and their applications.
4. How to rationalize denominators involving radicals.
5. Exploring the properties of cube roots and their geometric interpretations.

**Tip:** When dealing with radicals and exponents, always try to express everything using exponents first, as it often makes simplification easier!

Learn how to introduce factors under radicals and simplify expressions step-by-step. This detailed explanation covers key mathematical concepts like radicals, exponents, and cube roots. Suitable for high school students and anyone interested in algebraic simplification.

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