Certainly! Let's break down and simplify the expression [tex]\(2 \sqrt[3]{27 x^3 y^6}\)[/tex] step-by-step.
### Step 1: Simplification Inside the Cube Root
First, we need to simplify the expression inside the cube root, [tex]\( \sqrt[3]{27 x^3 y^6} \)[/tex].
1. Cube Root of 27:
- We know that [tex]\( 27 = 3^3 \)[/tex]. Therefore, [tex]\( \sqrt[3]{27} = 3 \)[/tex] since [tex]\( 3^3 = 27 \)[/tex].
2. Cube Root of [tex]\( x^3 \)[/tex]:
- The cube root of [tex]\( x^3 \)[/tex] is [tex]\( x \)[/tex] because [tex]\( (x^3)^{1/3} = x \)[/tex].
3. Cube Root of [tex]\( y^6 \)[/tex]:
- The cube root of [tex]\( y^6 \)[/tex] can be simplified as follows: [tex]\( (y^6)^{1/3} = y^{6/3} = y^2 \)[/tex].
Combining these results, we have:
[tex]\[ \sqrt[3]{27 x^3 y^6} = 3 \cdot x \cdot y^2 \][/tex]
### Step 2: Multiply by 2
Next, we need to multiply the simplified result inside the cube root by 2:
[tex]\[ 2 \cdot \sqrt[3]{27 x^3 y^6} = 2 \cdot (3 \cdot x \cdot y^2) \][/tex]
Perform the multiplication:
[tex]\[ 2 \cdot 3 \cdot x \cdot y^2 = 6 x y^2 \][/tex]
### Final Answer
Thus, the simplified form of the expression [tex]\( 2 \sqrt[3]{27 x^3 y^6} \)[/tex] is:
[tex]\[ 6 x y^2 \][/tex]
So, the answer is:
[tex]\[ 6 x y^2 \][/tex]
