Simplify the expression $3x^2 \cdot (2x^2)^3$.

1. Problem Description

Simplify the expression $3x^2 \cdot (2x^2)^3$.

2. Solution Steps

First, we need to simplify the term $(2x^2)^3$. Using the power of a product rule, we have:

$(ab)^n = a^n b^n$

Therefore, $(2x^2)^3 = 2^3 (x^2)^3 = 8(x^2)^3$.

Next, we simplify $(x^2)^3$ using the power of a power rule:

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

So, $(x^2)^3 = x^{2 \cdot 3} = x^6$.

Thus, $(2x^2)^3 = 8x^6$.

Now we substitute this back into the original expression:

$3x^2 \cdot (2x^2)^3 = 3x^2 \cdot 8x^6$.

Multiply the coefficients and use the product of powers rule $a^m \cdot a^n = a^{m+n}$ to simplify the variable terms:

$3x^2 \cdot 8x^6 = (3 \cdot 8) \cdot (x^2 \cdot x^6) = 24x^{2+6} = 24x^8$.

3. Final Answer

The simplified expression is $24x^8$.