(2a^3)(2h^4)^3[(-2g^4h)^3]^2

Steps to Solve

1. Expand the Expression
   First, we will expand each part of the expression:
   $$(2a^3) \cdot (2h^4)^3 \cdot [(-2g^4h)^3]^2$$

2. Calculate Each Term
   For $(2h^4)^3$, we apply the power rule:
   $$(2h^4)^3 = 2^3 \cdot (h^4)^3 = 8h^{12}$$
   For $(-2g^4h)^3$, we apply the same rules:
   $$(-2g^4h)^3 = (-2)^3 \cdot (g^4)^3 \cdot h^3 = -8g^{12}h^3$$

3. Square the Result
   Now we need to square the result from the last term:
   $$[(-2g^4h)^3]^2 = (-8g^{12}h^3)^2 = 64g^{24}h^6$$

4. Combine All Terms
   Now substitute the expanded results into the expression:
   $$(2a^3) \cdot (8h^{12}) \cdot (64g^{24}h^6)$$

5. Calculate the Coefficients
   Combine the numbers (coefficients) together:
   $$2 \cdot 8 \cdot 64 = 1024$$

6. Combine the Variables
   Now combine the variable parts:

• For $h$: $h^{12} \cdot h^6 = h^{12+6} = h^{18}$
• The rest stays the same.

The combined final result is: $$1024 a^3 g^{24} h^{18}$$