Sure, let's simplify each expression step-by-step.
### Simplifying [tex]\((a^3 b^2)^4\)[/tex]
1. Expand the powers in the expression:
[tex]\[
(a^3 b^2)^4
\][/tex]
2. Apply the power rule for exponents [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[
(a^3)^4 \cdot (b^2)^4
\][/tex]
3. Calculate the exponents separately:
[tex]\[
a^{3 \cdot 4} \cdot b^{2 \cdot 4}
\][/tex]
4. Multiply the exponents:
[tex]\[
a^{12} \cdot b^{8}
\][/tex]
Thus, the simplified form of [tex]\((a^3 b^2)^4\)[/tex] is:
[tex]\[
a^{12} b^8
\][/tex]
### Simplifying [tex]\((2 x^2 y)^3 \cdot (4 + y^3)^2\)[/tex]
1. Expand the first part [tex]\((2 x^2 y)^3\)[/tex]:
[tex]\[
(2 x^2 y)^3
\][/tex]
2. Apply the power rule for exponents:
[tex]\[
2^3 \cdot (x^2)^3 \cdot y^3
\][/tex]
3. Calculate the exponents separately and the constant power:
[tex]\[
8 \cdot x^{2 \cdot 3} \cdot y^3
\][/tex]
4. Multiply the exponents:
[tex]\[
8 \cdot x^6 \cdot y^3
\][/tex]
5. Now expand the second part [tex]\((4 + y^3)^2\)[/tex]:
[tex]\[
(4 + y^3)^2
\][/tex]
6. We treat this as a binomial expression and square it directly as it stands.
Thus, we have:
[tex]\[
(8 x^6 y^3) \cdot (4 + y^3)^2
\][/tex]
So, the final simplified form of [tex]\((2 x^2 y)^3 \cdot (4 + y^3)^2\)[/tex] is:
[tex]\[
8 x^6 y^3 (4 + y^3)^2
\][/tex]
Thus, the complete simplified expressions are:
[tex]\[
\begin{array}{l}
(a^3 b^2)^4 = a^{12} b^8, \\
(2 x^2 y)^3 \cdot (4 + y^3)^2 = 8 x^6 y^3 (4 + y^3)^2
\end{array}
\][/tex]
