Let's start by analyzing and simplifying the expression [tex]\(\left(8 x^4 y^3\right)^2\)[/tex].
Step 1: Apply the power of a product rule.
The power of a product rule states that [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]. In this case, we have three factors inside the parentheses: the coefficient 8, [tex]\(x^4\)[/tex], and [tex]\(y^3\)[/tex]. We need to raise each of these factors to the power of 2.
Thus, [tex]\(\left(8 x^4 y^3\right)^2\)[/tex] becomes:
[tex]\[ (8)^2 \cdot (x^4)^2 \cdot (y^3)^2 \][/tex]
Step 2: Simplify each term separately.
1. Simplify [tex]\((8)^2\)[/tex]:
[tex]\[ (8)^2 = 8 \times 8 = 64 \][/tex]
2. Simplify [tex]\((x^4)^2\)[/tex]:
[tex]\[ (x^4)^2 = x^{4 \cdot 2} = x^8 \][/tex]
3. Simplify [tex]\((y^3)^2\)[/tex]:
[tex]\[ (y^3)^2 = y^{3 \cdot 2} = y^6 \][/tex]
Step 3: Combine the simplified terms.
Putting it all together, we get:
[tex]\[ (8)^2 \cdot (x^4)^2 \cdot (y^3)^2 = 64 \cdot x^8 \cdot y^6 \][/tex]
Therefore, the correct simplification of [tex]\(\left(8 x^4 y^3\right)^2\)[/tex] is:
[tex]\[ 64 x^8 y^6 \][/tex]
Among the given options, the correct one is:
[tex]\[ \boxed{64 x^8 y^6} \][/tex]
