https://file.aiquickdraw.com/mathbot/file/1726846032860me1jlj2a.jpg

Let's go through each of the expressions one by one:

### 10. \( (m^2 v)^3 \)
We apply the power of a power rule:

\[
(m^2 v)^3 = (m^2)^3 \cdot v^3 = m^{2 \cdot 3} \cdot v^3 = m^6 v^3
\]

### 11. \( \frac{w^7}{w^{12}} \)
We apply the quotient rule for exponents: \( \frac{w^a}{w^b} = w^{a-b} \).

\[
\frac{w^7}{w^{12}} = w^{7-12} = w^{-5}
\]

### 12. \( (-2x^2 y^3)^3 \)
We apply the power of a product rule: \( (abc)^n = a^n b^n c^n \).

\[
(-2x^2 y^3)^3 = (-2)^3 \cdot (x^2)^3 \cdot (y^3)^3
\]
Calculating each term:
\[
(-2)^3 = -8, \quad (x^2)^3 = x^6, \quad (y^3)^3 = y^9
\]
So the result is:
\[
-8x^6 y^9
\]

Let me know if you want more details or have any questions. Here are some related questions:

1. How do you simplify expressions with fractional exponents?
2. What are the key properties of negative exponents?
3. How do you handle exponents when dividing variables with the same base?
4. What is the general formula for raising a product to a power?
5. Can you explain how exponents behave when they involve negative numbers?

**Tip:** Always group like terms and apply the appropriate exponent rules to simplify any expressions efficiently.

Learn how to simplify expressions involving exponents such as (m^2 v)^3, w^7/w^12, and (-2x^2 y^3)^3. This guide covers key exponent rules, including power of a product and the quotient rule. Suitable for students in Grades 8-10.

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