Answer :

Let's simplify the given expression step-by-step.

We start with the expression inside the square root:

[tex]\[ 3 x^2 \cdot 5 y^2 \][/tex]

First, combine the constants and the variables separately:

[tex]\[ 3 \cdot 5 \cdot x^2 \cdot y^2 \][/tex]

Multiplying the constants:

[tex]\[ 15 \cdot x^2 \cdot y^2 \][/tex]

Now, we need to find the square root of this product:

[tex]\[ \sqrt{15 x^2 y^2} \][/tex]

We can use the property of square roots that states:

[tex]\[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \][/tex]

So, let's apply this property to separate the square root of the product:

[tex]\[ \sqrt{15 x^2 y^2} = \sqrt{15} \cdot \sqrt{x^2 y^2} \][/tex]

Next, we simplify [tex]\(\sqrt{x^2 y^2}\)[/tex]:

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

Since the square root of a square cancels out, this becomes:

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

Now, putting it all together:

[tex]\[ \sqrt{15 x^2 y^2} = \sqrt{15} \cdot x \cdot y \][/tex]

Thus, the simplified form of the given expression is:

[tex]\[ \sqrt{3 x^2 \cdot 5 y^2} = \sqrt{15} \cdot x \cdot y \][/tex]

Therefore:

[tex]\[ \boxed{\sqrt{15} \cdot x \cdot y} \][/tex]