simplify using only positive exponents \left(m^3\right)^4\cdot 2m^4\cdot m^2

To simplify the expression \(\left(m^3\right)^4 \cdot 2m^4 \cdot m^2\) using only positive exponents, follow these steps:

1. **Apply the power rule to \(\left(m^3\right)^4\):**
   \[
   \left(m^3\right)^4 = m^{3 \cdot 4} = m^{12}
   \]
   The power rule states that \(\left(a^m\right)^n = a^{mn}\).

2. **Combine all the \(m\) terms:**
   \[
   m^{12} \cdot 2m^4 \cdot m^2
   \]
   This can be rewritten as:
   \[
   2 \cdot m^{12 + 4 + 2}
   \]

3. **Add the exponents:**
   \[
   2 \cdot m^{18}
   \]

The simplified expression is:
\[
2m^{18}
\]

Would you like any further details or have any questions?

Here are 8 related questions to expand on this:

1. How do you simplify expressions with multiple exponents and coefficients?
2. Can you explain the power rule in more detail?
3. How would the simplification process change if there were negative exponents involved?
4. What if the expression included division instead of multiplication?
5. How do you handle expressions with different variables in addition to exponents?
6. Can you provide an example where the base is a fraction instead of a variable?
7. What is the importance of keeping exponents positive in simplified expressions?
8. How would you simplify an expression like \(\left(m^2n^3\right)^2 \cdot m^4n^{-2}\)?

**Tip:** When dealing with multiple terms with the same base, always add the exponents for multiplication and subtract them for division.

Learn how to simplify the expression (m^3)^4 * 2m^4 * m^2 using positive exponents. Follow a detailed step-by-step solution covering the power rule and multiplication of exponents. Suitable for high school students and anyone interested in algebraic simplification.

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