To solve the problem of finding which expression is equivalent to [tex]\(\left(4 y^2\right)^3 \left(3 y^2\right)\)[/tex], let's go through the steps for simplifying the given expression.
1. Simplify [tex]\(\left(4 y^2\right)^3\)[/tex]:
[tex]\[
\left(4 y^2\right)^3 = 4^3 \cdot (y^2)^3
\][/tex]
We know that [tex]\(a^3 = a \cdot a \cdot a\)[/tex], so:
[tex]\[
4^3 = 64
\][/tex]
Also, [tex]\((y^2)^3 = y^{2 \cdot 3} = y^6\)[/tex]:
[tex]\[
\left(4 y^2\right)^3 = 64 y^6
\][/tex]
2. Multiply the result by [tex]\(3 y^2\)[/tex]:
[tex]\[
(64 y^6) \cdot (3 y^2)
\][/tex]
Combine the coefficients (numeric parts):
[tex]\[
64 \cdot 3 = 192
\][/tex]
Combine the variables with exponents using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
y^6 \cdot y^2 = y^{6+2} = y^8
\][/tex]
3. Put everything together:
[tex]\[
192 y^8
\][/tex]
Thus, the expression that is equivalent to [tex]\(\left(4 y^2\right)^3 \left(3 y^2\right)\)[/tex] is:
[tex]\[
\boxed{192 y^8}
\][/tex]
