To simplify the expression $(5m^2n^7)^3$, you need to apply the power of a power rule, which states:

$(a^m)^n = a^{m \cdot n}$

Here’s how we apply that to each part of the expression:

$(5m^2n^7)^3 = 5^3 \cdot (m^2)^3 \cdot (n^7)^3$

Now simplify each part:

1. $5^3 = 125$
2. $(m^2)^3 = m^{2 \cdot 3} = m^6$
3. $(n^7)^3 = n^{7 \cdot 3} = n^{21}$

Thus, the simplified expression is:

$125m^6n^{21}$

Answer: $125m^6n^{21}$

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Would you like more details on how I applied the power rule or other steps?

Related Questions:

1. How do you simplify expressions with exponents when you multiply or divide terms?
2. What is the rule for raising a product to a power, and how does it differ from raising a power to a power?
3. How do you handle expressions with negative exponents?
4. How can you apply the distributive property when working with exponents?
5. What is the rule for simplifying fractional exponents?

Tip: When dealing with powers, always ensure that you multiply the exponents when raising a power to a power, and use the distributive property when raising a product to a power.