Sure, let's work through the simplification of the given expression step-by-step.
We are given the expression:
[tex]\[
\sqrt{\frac{x^4 y}{225}}
\][/tex]
First, recognize that the square root of a quotient can be written as the quotient of the square roots:
[tex]\[
\sqrt{\frac{x^4 y}{225}} = \frac{\sqrt{x^4 y}}{\sqrt{225}}
\][/tex]
Next, let's simplify each part of the quotient separately.
1. Simplify the denominator:
[tex]\[
\sqrt{225}
\][/tex]
Since 225 is a perfect square:
[tex]\[
\sqrt{225} = 15
\][/tex]
2. Now, simplify the numerator:
[tex]\[
\sqrt{x^4 y}
\][/tex]
Notice that [tex]\(\sqrt{x^4 y}\)[/tex] can be broken into the product of square roots:
[tex]\[
\sqrt{x^4 y} = \sqrt{x^4} \cdot \sqrt{y}
\][/tex]
Since [tex]\(x^4\)[/tex] is a perfect square ([tex]\(x^4 = (x^2)^2\)[/tex]):
[tex]\[
\sqrt{x^4} = x^2
\][/tex]
Then the expression becomes:
[tex]\[
\sqrt{x^4} \cdot \sqrt{y} = x^2 \cdot \sqrt{y}
\][/tex]
Putting it all together, we get:
[tex]\[
\frac{\sqrt{x^4 y}}{\sqrt{225}} = \frac{x^2 \cdot \sqrt{y}}{15}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{x^2 \sqrt{y}}{15}
\][/tex]
Or equivalently, written without the fraction and as a single fraction:
[tex]\[
\frac{\sqrt{x^4 y}}{15} = \frac{x^2 \sqrt{y}}{15}
\][/tex]
Therefore, the final simplified expression is:
[tex]\[
\frac{x^2 \sqrt{y}}{15}
\][/tex]
This demonstrates that the simplified result of [tex]\(\sqrt{\frac{x^4 y}{225}}\)[/tex] is [tex]\(\frac{x^2 \sqrt{y}}{15}\)[/tex].
