Certainly! Let's break down the simplification of the given expression step-by-step.
The given expression is:
[tex]\[ \sqrt{\frac{x y^2}{16}} \][/tex]
### Step 1: Simplify the square root of a fraction
The square root of a fraction can be expressed as the square root of the numerator divided by the square root of the denominator. Therefore, we can rewrite the expression as:
[tex]\[ \sqrt{\frac{x y^2}{16}} = \frac{\sqrt{x y^2}}{\sqrt{16}} \][/tex]
### Step 2: Simplify the denominator
The square root of 16 is 4, so we can simplify the denominator:
[tex]\[ \frac{\sqrt{x y^2}}{\sqrt{16}} = \frac{\sqrt{x y^2}}{4} \][/tex]
### Step 3: Simplify the square root in the numerator
Next, consider the square root in the numerator. The expression [tex]\(\sqrt{x y^2}\)[/tex] can be broken down because the square root of a product is the product of the square roots. So:
[tex]\[ \sqrt{x y^2} = \sqrt{x} \cdot \sqrt{y^2} \][/tex]
We know that [tex]\(\sqrt{y^2}\)[/tex] is simply [tex]\(y\)[/tex] because squaring and then taking the square root of a variable returns the variable itself:
[tex]\[ \sqrt{x} \cdot \sqrt{y^2} = \sqrt{x} \cdot y \][/tex]
### Step 4: Combine the simplified parts
Now, substitute this back into our fraction:
[tex]\[ \frac{\sqrt{x} \cdot y}{4} \][/tex]
### Final Simplified Expression
Thus, the simplified form of the given expression [tex]\(\sqrt{\frac{x y^2}{16}}\)[/tex] is:
[tex]\[ \frac{\sqrt{x} \cdot y}{4} \][/tex]
In another form, it can also be written as:
[tex]\[ \frac{y \sqrt{x}}{4} \][/tex]
So the final simplified expression is:
[tex]\[ \sqrt{\frac{x y^2}{16}} = \frac{y \sqrt{x}}{4} \][/tex]
