To determine the equivalent expression for [tex]\(\left(2 x^4 y\right)^3\)[/tex], let's break the problem down step-by-step using exponent rules.
We start with the expression:
[tex]\[
\left(2 x^4 y\right)^3
\][/tex]
### Step 1: Apply the Power to Each Factor
When raising a product to a power, we apply the exponent to each factor within the parentheses. This means that we distribute the exponent of 3 to each part of the product:
[tex]\[
\left(2\right)^3 \left(x^4\right)^3 \left(y\right)^3
\][/tex]
### Step 2: Simplify Each Term
Now, let's simplify each term individually.
1. Simplify [tex]\((2)^3\)[/tex]:
[tex]\[
2^3 = 8
\][/tex]
2. Simplify [tex]\((x^4)^3\)[/tex]:
[tex]\[
(x^4)^3 = x^{4 \cdot 3} = x^{12}
\][/tex]
3. Simplify [tex]\((y)^3\)[/tex]:
[tex]\[
(y)^3 = y^{1 \cdot 3} = y^3
\][/tex]
### Step 3: Combine the Results
Putting all the simplified terms together, we get:
[tex]\[
\left(2 x^4 y\right)^3 = 8 x^{12} y^3
\][/tex]
Therefore, the expression equivalent to [tex]\(\left(2 x^4 y\right)^3\)[/tex] is:
[tex]\[
\boxed{8 x^{12} y^3}
\][/tex]
