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Simplify the radical. Assume that all variables represent positive real numbers.

[tex]\[ \sqrt[3]{-64 x^{12} y^6} \][/tex]


Sagot :

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]