IDNLearn.com: Where your questions meet expert advice and community insights. Join our Q&A platform to access reliable and detailed answers from experts in various fields.
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]
### 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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.