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Which expression is equivalent to [tex]\left(2 x^4 y\right)^3[/tex]?

A. [tex]2 x^7 y^4[/tex]

B. [tex]2 x^{12} y^3[/tex]

C. [tex]8 x^7 y^4[/tex]

D. [tex]8 x^{12} y^3[/tex]


Sagot :

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]