To simplify the expression \((2m^5)^6\), follow these steps:

### Step 1: Apply the Power of a Product Rule
The power of a product rule states that \((ab)^n = a^n \cdot b^n\). In this case, you can apply it to both the constant \(2\) and the variable term \(m^5\):

\[
(2m^5)^6 = 2^6 \cdot (m^5)^6
\]

### Step 2: Simplify Each Term
- \(2^6 = 64\)
- For \((m^5)^6\), use the power of a power rule, which states \((x^a)^b = x^{a \cdot b}\). So, \((m^5)^6 = m^{5 \cdot 6} = m^{30}\).

### Step 3: Write the Final Expression
Now combine everything:

\[
(2m^5)^6 = 64m^{30}
\]

Thus, the simplified expression is:

\[
\boxed{64m^{30}}
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand on this:

1. How would you simplify \((3x^2)^4\)?
2. What happens when simplifying \((5y^3)^2\)?
3. How does the power rule work when applying it to a negative exponent, e.g., \((2m^{-3})^4\)?
4. How do you simplify a product of two powers, e.g., \((2m^2)^3 \cdot (4m^4)^2\)?
5. Can you explain how the product rule is applied to expressions with multiple terms?

**Tip:** When raising a product to a power, apply the exponent to each factor inside the parentheses.

Learn how to simplify the expression (2m^5)^6 using exponent rules. This problem demonstrates the power of a product rule and power of a power rule to simplify terms step by step. Suitable for students in Grades 8-10.

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