To determine which expression is equivalent to [tex]\(\left(4 y^2\right)^3 \left(3 y^2\right)\)[/tex], let's work through it step by step:
1. Simplify [tex]\((4 y^2)^3\)[/tex]:
[tex]\((4 y^2)^3\)[/tex] means raising [tex]\(4 y^2\)[/tex] to the power of 3. This can be represented as:
[tex]\[
4^3 \cdot (y^2)^3.
\][/tex]
- [tex]\(4^3 = 4 \times 4 \times 4 = 64\)[/tex].
- [tex]\((y^2)^3 = y^{2 \times 3} = y^6\)[/tex].
Therefore, [tex]\((4 y^2)^3\)[/tex] simplifies to:
[tex]\[
64 y^6.
\][/tex]
2. Simplify [tex]\(3 y^2\)[/tex]:
Since there is no exponent to multiply, it remains the same:
[tex]\[
3 y^2.
\][/tex]
3. Multiply the simplified expressions together:
Now we multiply [tex]\(64 y^6\)[/tex] by [tex]\(3 y^2\)[/tex]:
[tex]\[
(64 y^6) \cdot (3 y^2).
\][/tex]
- Multiply the coefficients [tex]\(64\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
64 \times 3 = 192.
\][/tex]
- Multiply the variable parts [tex]\(y^6\)[/tex] and [tex]\(y^2\)[/tex]:
[tex]\[
y^6 \cdot y^2 = y^{6+2} = y^8.
\][/tex]
4. Combine the results:
Putting it all together, the expression [tex]\(\left(4 y^2\right)^3 \left(3 y^2\right)\)[/tex] simplifies to:
[tex]\[
192 y^8.
\][/tex]
Hence, the equivalent expression is [tex]\(\boxed{192 y^8}\)[/tex].
