Alright, let's simplify the given expression step by step:
Given:
[tex]\[ \sqrt[3]{686 x^4 y^7} \][/tex]
### Step 1: Factorize the constant 686
686 can be factorized as:
[tex]\[ 686 = 7 \times 98 = 7 \times 7 \times 14 = 7 \times 7 \times 7 \times 2 = 7^3 \times 2 \][/tex]
So, we have:
[tex]\[ 686 = 7^3 \times 2 \][/tex]
### Step 2: Simplify the variables within the cube root
For the variable \( x^4 \), we can rewrite it as:
[tex]\[ x^4 = x^3 \times x \][/tex]
The cube root of \( x^3 \) is \( x \), and \( x \) remains inside the radical:
[tex]\[ \sqrt[3]{x^3 \times x} = x \cdot \sqrt[3]{x} \][/tex]
For the variable \( y^7 \), we can rewrite it as:
[tex]\[ y^7 = y^6 \times y \][/tex]
The cube root of \( y^6 \) is \( y^2 \), and \( y \) remains inside the radical:
[tex]\[ \sqrt[3]{y^6 \times y} = y^2 \cdot \sqrt[3]{y} \][/tex]
### Step 3: Combine the simplified results
We have:
[tex]\[ \sqrt[3]{686 x^4 y^7} = \sqrt[3]{7^3 \times 2 \times x^3 \times x \times y^6 \times y} \][/tex]
This can be separated into:
[tex]\[ 7 \cdot x \cdot y^2 \cdot \sqrt[3]{2 \times x \times y} \][/tex]
So, we have:
[tex]\[ \sqrt[3]{686 x^4 y^7} = 7 x y^2 \cdot \sqrt[3]{2 x y} \][/tex]
### Final Answer
So, the simplified form of the given expression is:
[tex]\[ 7 x y^2 \sqrt[3]{2 x y} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
