To determine the cube root of [tex]\(\sqrt[3]{64 a^{12}}\)[/tex], we need to break down and simplify the expression step-by-step.
First, let's look at each component inside the cube root.
### Step 1: Simplify the constant 64
64 can be written as a power of 4:
[tex]\[ 64 = 4^3 \][/tex]
### Step 2: Simplify [tex]\(a^{12}\)[/tex]
The exponent 12 can be grouped as a power of 4:
[tex]\[ a^{12} = (a^4)^3 \][/tex]
### Step 3: Combine the expressions inside the cube root
Now we can write the expression inside the cube root as:
[tex]\[ \sqrt[3]{64 a^{12}} = \sqrt[3]{4^3 \cdot (a^4)^3} \][/tex]
### Step 4: Apply the cube root to the entire expression
When we apply the cube root to both components:
[tex]\[ \sqrt[3]{4^3} \times \sqrt[3]{(a^4)^3} \][/tex]
Since taking the cube root of [tex]\(4^3\)[/tex] gives 4, and the cube root of [tex]\((a^4)^3\)[/tex] is [tex]\(a^4\)[/tex]:
[tex]\[ \sqrt[3]{4^3} = 4 \][/tex]
[tex]\[ \sqrt[3]{(a^4)^3} = a^4 \][/tex]
### Step 5: Multiply the simplified components
Putting it all together, we obtain:
[tex]\[ \sqrt[3]{64 a^{12}} = 4 \cdot a^4 = 4a^4 \][/tex]
Thus, the simplified form of [tex]\(\sqrt[3]{64 a^{12}}\)[/tex] is:
[tex]\[ \boxed{4 a^4} \][/tex]
So the correct answer is:
[tex]\[ 4 a^4 \][/tex]
