To simplify the radical expression [tex]\(\sqrt[3]{-64 x^{12} y^6}\)[/tex], we will break down the expression and apply the properties of radicals and exponents step-by-step.
### Step 1: Recognize the Cube Root
The given expression is a cube root:
[tex]\[
\sqrt[3]{-64 x^{12} y^6}
\][/tex]
### Step 2: Break Down Each Factor Inside the Radical
We need to separate and simplify each component inside the cube root individually:
[tex]\[
\sqrt[3]{-64 x^{12} y^6} = \sqrt[3]{-64} \cdot \sqrt[3]{x^{12}} \cdot \sqrt[3]{y^6}
\][/tex]
### Step 3: Simplify Each Cube Root
#### Simplify [tex]\(\sqrt[3]{-64}\)[/tex]:
[tex]\[
\sqrt[3]{-64} = -\sqrt[3]{64} = -4
\][/tex]
This is true because [tex]\( (-4)^3 = -64 \)[/tex].
#### Simplify [tex]\(\sqrt[3]{x^{12}}\)[/tex]:
[tex]\[
\sqrt[3]{x^{12}} = x^{12/3} = x^4
\][/tex]
#### Simplify [tex]\(\sqrt[3]{y^6}\)[/tex]:
[tex]\[
\sqrt[3]{y^6} = y^{6/3} = y^2
\][/tex]
### Step 4: Combine the Results
Multiplying the simplified parts together:
[tex]\[
-4 \cdot x^4 \cdot y^2 = -4x^4y^2
\][/tex]
### Final Answer
The simplified form of the radical expression is:
[tex]\[
\boxed{-4x^4y^2}
\][/tex]
