We are given the expression $a(b^2 \cdot c \cdot d^2)$ and the values:

• $a = -1$
• $b = 2$
• $c = -3$
• $d = 4$

Step-by-step Solution:

1. Substitute the values into the expression: $a(b^2 \cdot c \cdot d^2) = -1 \cdot (2^2 \cdot (-3) \cdot 4^2)$

2. Evaluate $b^2$: $b^2 = 2^2 = 4$

3. Evaluate $d^2$: $d^2 = 4^2 = 16$

4. Multiply $b^2$, $c$, and $d^2$: $4 \cdot (-3) \cdot 16 = -12 \cdot 16 = -192$

5. Multiply the result by $a$: $-1 \cdot (-192) = 192$

Final Answer:

$\boxed{192}$

Let me know if you'd like a detailed explanation of any step!

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Related Questions:

1. How does changing $b$ to $-2$ affect the result?
2. What happens if $c = 0$? Does the answer change?
3. How can we generalize this process for any values of $a, b, c, d$?
4. What is the importance of exponents in simplifying such expressions?
5. Can we solve a similar expression with fractional values for $b$ and $d$?

Tip: Always handle negative signs carefully in multiplication, especially when working with powers!