To simplify the expression [tex]\(\left(4 a^4 b\right)^2\)[/tex], follow these steps:
1. Identify the expression and the exponent: We are given [tex]\(\left(4 a^4 b\right)^2\)[/tex].
2. Distribute the external exponent to each term inside the parentheses: The expression inside the parentheses consists of three factors: [tex]\(4\)[/tex], [tex]\(a^4\)[/tex], and [tex]\(b\)[/tex]. When we raise the entire expression to the power of 2, we need to raise each factor inside the parentheses to the power of 2.
[tex]\[
\left(4 a^4 b\right)^2 = 4^2 \cdot (a^4)^2 \cdot b^2
\][/tex]
3. Simplify each part separately:
- For [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
- For [tex]\((a^4)^2\)[/tex], use the property of exponents that [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[
(a^4)^2 = a^{4 \cdot 2} = a^8
\][/tex]
- For [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = b^2
\][/tex]
4. Combine the results:
[tex]\[
4^2 \cdot (a^4)^2 \cdot b^2 = 16 \cdot a^8 \cdot b^2 = 16 a^8 b^2
\][/tex]
Therefore, the correct simplification of the expression [tex]\(\left(4 a^4 b\right)^2\)[/tex] is:
[tex]\[
\boxed{16 a^8 b^2}
\][/tex]
