To simplify the given expression:
[tex]\[ \sqrt{x^2 y} - 3 \sqrt{4 x^2 y} + \sqrt{9 x^2 y} \][/tex]
we will work through each term step-by-step and then combine them.
1. Simplify [tex]\(\sqrt{x^2 y}\)[/tex]:
- Recall that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Therefore:
[tex]\[ \sqrt{x^2 y} = \sqrt{x^2} \cdot \sqrt{y} \][/tex]
- Since [tex]\(x\)[/tex] is nonnegative, [tex]\(\sqrt{x^2} = x\)[/tex]. Hence:
[tex]\[ \sqrt{x^2 y} = x \sqrt{y} \][/tex]
2. Simplify [tex]\(3 \sqrt{4 x^2 y}\)[/tex]:
- Similarly:
[tex]\[ \sqrt{4 x^2 y} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y} \][/tex]
- We know that [tex]\(\sqrt{4} = 2\)[/tex] and [tex]\(\sqrt{x^2} = x\)[/tex]:
[tex]\[ \sqrt{4 x^2 y} = 2 x \sqrt{y} \][/tex]
- Now multiply this by 3:
[tex]\[ 3 \sqrt{4 x^2 y} = 3 \cdot 2 x \sqrt{y} = 6 x \sqrt{y} \][/tex]
3. Simplify [tex]\(\sqrt{9 x^2 y}\)[/tex]:
- Again, using the same approach:
[tex]\[ \sqrt{9 x^2 y} = \sqrt{9} \cdot \sqrt{x^2} \cdot \sqrt{y} \][/tex]
- Since [tex]\(\sqrt{9} = 3\)[/tex] and [tex]\(\sqrt{x^2} = x\)[/tex]:
[tex]\[ \sqrt{9 x^2 y} = 3 x \sqrt{y} \][/tex]
Now, substitute the simplified terms back into the original expression:
[tex]\[ x \sqrt{y} - 6 x \sqrt{y} + 3 x \sqrt{y} \][/tex]
Combine these terms:
[tex]\[ (1 - 6 + 3) x \sqrt{y} \][/tex]
[tex]\[ (-2) x \sqrt{y} \][/tex]
Thus, the simplified expression is:
[tex]\[ -2 x \sqrt{y} \][/tex]
Therefore, the answer is:
[tex]\[ -2 x \sqrt{y} \][/tex]
