Answer :

To simplify the given expression:

[tex]\[ y \sqrt{28 x^3} + 6 x \sqrt{7 x y^2} \][/tex]

we need to break it down into manageable parts.

### Step 1: Simplify Each Term Individually

#### Term 1: [tex]\( y \sqrt{28 x^3} \)[/tex]

1. Recognize that we can break the expression inside the square root into prime factors:
[tex]\[ \sqrt{28 x^3} = \sqrt{4 \cdot 7 \cdot x^3} \][/tex]
2. Simplify the square root of each factor where possible:
[tex]\[ \sqrt{4 \cdot 7 \cdot x^3} = \sqrt{4} \cdot \sqrt{7} \cdot \sqrt{x^3} = 2 \sqrt{7} \cdot x^{3/2} \][/tex]
3. Substitute this back into the original term:
[tex]\[ y \sqrt{28 x^3} = y \cdot 2 \sqrt{7} \cdot x^{3/2} = 2 y \sqrt{7} \cdot x^{3/2} \][/tex]

#### Term 2: [tex]\( 6 x \sqrt{7 x y^2} \)[/tex]

1. Again break down the expression inside the square root:
[tex]\[ \sqrt{7 x y^2} = \sqrt{7} \cdot \sqrt{x} \cdot \sqrt{y^2} \][/tex]
2. Simplify the square root of each factor where possible:
[tex]\[ \sqrt{7 x y^2} = \sqrt{7} \cdot \sqrt{x} \cdot y \][/tex]
3. Substitute this back into the original term:
[tex]\[ 6 x \sqrt{7 x y^2} = 6 x \cdot \sqrt{7} \cdot \sqrt{x} \cdot y = 6 y \sqrt{7} \cdot x \cdot \sqrt{x} = 6 y \sqrt{7} \cdot x^{3/2} \][/tex]

### Step 2: Combine the Simplified Terms

Now combining the two simplified terms together:

1. From Term 1:
[tex]\[ 2 y \sqrt{7} \cdot x^{3/2} \][/tex]

2. From Term 2:
[tex]\[ 6 y \sqrt{7} \cdot x^{3/2} \][/tex]

Combine them:
[tex]\[ 2 y \sqrt{7} \cdot x^{3/2} + 6 y \sqrt{7} \cdot x^{3/2} \][/tex]

### Step 3: Factor Out Common Factors

1. Both terms have a common factor of [tex]\( y \sqrt{7} x^{3/2} \)[/tex]:
[tex]\[ (2 + 6) y \sqrt{7} \cdot x^{3/2} \][/tex]
2. Simplify inside the parentheses:
[tex]\[ (2 + 6) y \sqrt{7} \cdot x^{3/2} = 8 y \sqrt{7} \cdot x^{3/2} \][/tex]

Therefore, the simplified form of the given expression is:

[tex]\[ 8 y \sqrt{7} x^{3/2} \][/tex]