Answer :

Let's break down and simplify the expression [tex]\(2 \sqrt{75 x y^3}\)[/tex] step by step.

1. Factorize the number inside the square root:

The number 75 can be factorized as [tex]\(75 = 25 \times 3\)[/tex].

2. Rewrite the square root expression:

[tex]\[ 2 \sqrt{75 x y^3} = 2 \sqrt{(25 \times 3) x y^3} \][/tex]

3. Separate the product inside the square root:

[tex]\[ 2 \sqrt{25 \times 3 x y^3} = 2 \sqrt{25} \cdot \sqrt{3 x y^3} \][/tex]

4. Simplify the square roots:

- [tex]\(\sqrt{25}\)[/tex] is 5.

Therefore,

[tex]\[ 2 \sqrt{25} \cdot \sqrt{3 x y^3} = 2 \times 5 \cdot \sqrt{3 x y^3} = 10 \sqrt{3 x y^3} \][/tex]

5. Simplify the square root involving the variables:

Note that [tex]\(y^3\)[/tex] can be written as [tex]\(y^2 \times y\)[/tex]. Thus,

[tex]\[ \sqrt{3 x y^3} = \sqrt{3 x y^2 \cdot y} = \sqrt{3 x y^2} \cdot \sqrt{y} \][/tex]

- [tex]\(\sqrt{y^2}\)[/tex] is [tex]\(y\)[/tex].

Therefore,

[tex]\[ \sqrt{3 x y^2 \cdot y} = \sqrt{3 x} \cdot y \cdot \sqrt{y} = y \cdot \sqrt{3 x y} \][/tex]

6. Combine the results:

Plugging this back into our simplified expression, we get:

[tex]\[ 10 \sqrt{3 x y^3} = 10 y \sqrt{3 x y} \][/tex]

Thus, the simplified form of [tex]\(2 \sqrt{75 x y^3}\)[/tex] is:

[tex]\[ 10 y \sqrt{3 x y} \][/tex]