To simplify the expression [tex]\(\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}\)[/tex], we will proceed step by step, making use of properties of square roots and simplifying the fractions where possible.
Step 1: Simplify the numerator.
The numerator is [tex]\(\sqrt{25 x^2 y^2}\)[/tex]. We can simplify this expression by recognizing that the square root of a product is the product of the square roots:
[tex]\[
\sqrt{25 x^2 y^2} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{y^2}.
\][/tex]
We know that:
[tex]\[
\sqrt{25} = 5, \quad \sqrt{x^2} = x \quad (\text{since } x \geq 0), \quad \text{and} \quad \sqrt{y^2} = y.
\][/tex]
So the numerator becomes:
[tex]\[
5 x y.
\][/tex]
Step 2: Simplify the denominator.
The denominator is [tex]\(\sqrt{x y}\)[/tex]. There is no further simplification needed for this part as it is already in its simplest form.
Step 3: Divide the simplified numerator by the simplified denominator.
We now have:
[tex]\[
\frac{5 x y}{\sqrt{x y}}.
\][/tex]
We can simplify this fraction by noting that [tex]\(\sqrt{x y}\)[/tex] is equivalent to [tex]\((x y)^{1/2}\)[/tex], and dividing powers of the same base involves subtracting the exponents. So we rewrite [tex]\(x y\)[/tex] as [tex]\( (x y)^{1/2}\cdot (x y)^{1/2}\)[/tex]:
[tex]\[
5 x y \cdot \frac{1}{(x y)^{1/2}} = 5 (x y)^{1 - \frac{1}{2}} = 5 (x y)^{\frac{1}{2}}.
\][/tex]
Since [tex]\((x y)^{\frac{1}{2}} = \sqrt{x y}\)[/tex], we can rewrite this as:
[tex]\[
5 \sqrt{x y}.
\][/tex]
Thus, the simplest form of [tex]\(\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}\)[/tex] is:
[tex]\[
\boxed{5 \sqrt{x y}}.
\][/tex]
