4x 
−3
 y 
2
 
(2x 
2
 y 
3
 ) 
4
 
​
  

Let's simplify the given algebraic expression step by step:

\[
4x^{-3} y^2 \cdot (2x^2 y^3)^4
\]

### Step 1: Expand the Power  
Using the **power rule** \((a^m)^n = a^{m \cdot n}\), expand the term inside the parentheses:

\[
(2x^2 y^3)^4 = 2^4 \cdot x^{2 \cdot 4} \cdot y^{3 \cdot 4}
\]

\[
= 16x^8 y^{12}
\]

### Step 2: Multiply with the First Term  
Now multiply:

\[
4x^{-3} y^2 \cdot 16x^8 y^{12}
\]

Multiply the constants:  
\[
4 \times 16 = 64
\]

Apply the **product rule** \((a^m \cdot a^n = a^{m+n})\):

\[
x^{-3} \cdot x^8 = x^{(-3 + 8)} = x^5
\]

\[
y^2 \cdot y^{12} = y^{(2+12)} = y^{14}
\]

### Final Answer:  
\[
\mathbf{64x^5 y^{14}}
\]

---

Would you like a more detailed explanation or have any questions?  

### Related Questions:
1. How do you simplify expressions with negative exponents?
2. What are the key exponent rules and their applications?
3. How do you handle fractional exponents in expressions?
4. Can you explain the difference between power and product rules?
5. How do you factor expressions with exponents efficiently?

### Tip:  
When dealing with exponents, always break down each step carefully and apply exponent rules systematically to avoid errors.

Learn how to simplify the expression 4x^-3 y^2 (2x^2 y^3)^4 using exponent rules like the power and product rules. This step-by-step solution helps clarify key algebraic concepts. Suitable for students in Grades 8-10.

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