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]
