To solve this question, we will separately simplify each provided mathematical expression, then combine the simplified results.
### Expression 1
[tex]\[
\sqrt{x^2 y^3} + 2 \sqrt{x^3 y^4} + x y \sqrt{y}
\][/tex]
First, we simplify each term individually:
1. [tex]\(\sqrt{x^2 y^3}\)[/tex]:
[tex]\[
\sqrt{x^2 y^3} = x y^{3/2}
\][/tex]
2. [tex]\(2 \sqrt{x^3 y^4}\)[/tex]:
[tex]\[
2 \sqrt{x^3 y^4} = 2 x^{3/2} y^2
\][/tex]
3. [tex]\(x y \sqrt{y}\)[/tex]:
[tex]\[
x y \sqrt{y} = x y \cdot y^{1/2} = x y^{3/2}
\][/tex]
Combining these results:
[tex]\[
\sqrt{x^2 y^3} + 2 \sqrt{x^3 y^4} + x y \sqrt{y} = x y^{3/2} + 2 x^{3/2} y^2 + x y^{3/2}
\][/tex]
Combine like terms:
[tex]\[
2 x y^{3/2} + 2 x^{3/2} y^2
\][/tex]
### Expression 2
[tex]\[
x^2 y^2 \sqrt{y} - 2 x y^2 \sqrt{x}
\][/tex]
We simplify each term individually:
1. [tex]\(x^2 y^2 \sqrt{y}\)[/tex]:
[tex]\[
x^2 y^2 \sqrt{y} = x^2 y^{5/2}
\][/tex]
2. [tex]\(-2 x y^2 \sqrt{x}\)[/tex]:
[tex]\[
-2 x y^2 \sqrt{x} = -2 x^{3/2} y^2
\][/tex]
Combine these results:
[tex]\[
x^2 y^{5/2} - 2 x^{3/2} y^2
\][/tex]
### Expression 3
[tex]\[
2 x y \sqrt{y} + 2 x y^2 \sqrt{x}
\][/tex]
We simplify each term individually:
1. [tex]\(2 x y \sqrt{y}\)[/tex]:
[tex]\[
2 x y \sqrt{y} = 2 x y^{3/2}
\][/tex]
2. [tex]\(2 x y^2 \sqrt{x}\)[/tex]:
[tex]\[
2 x y^2 \sqrt{x} = 2 x^{3/2} y^2
\][/tex]
Combine these results:
[tex]\[
2 x y^{3/2} + 2 x^{3/2} y^2
\][/tex]
### Expression 4
[tex]\[
4 x y \sqrt{x^5 y^8}
\][/tex]
We simplify the term:
[tex]\[
4 x y \sqrt{x^5 y^8} = 4 x y \cdot x^{5/2} y^4 = 4 x^{7/2} y^5
\][/tex]
### Expression 5
[tex]\[
2 x y \sqrt{x y}
\][/tex]
We simplify the term:
[tex]\[
2 x y \sqrt{x y} = 2 x y \cdot \sqrt{x} \cdot \sqrt{y} = 2 x y \cdot x^{1/2} y^{1/2} = 2 x^{3/2} y^{3/2}
\][/tex]
### Combining All Simplified Expressions
We sum the simplified results of each expression:
[tex]\[
(2 x^{3/2} y^2 + 2 x y^{3/2}) + (x^2 y^{5/2} - 2 x^{3/2} y^2) + (2 x^{3/2} y^2 + 2 x y^{3/2}) + 4 x^{7/2} y^5 + 2 x^{3/2} y^{3/2}
\][/tex]
Combining all like terms, we get:
[tex]\[
2 x y^{3/2} + x^2 y^{5/2} + 2 x y^{3/2} + 4 x^{7/2} y^5 + 2 x^{3/2} y^{3/2}
\][/tex]
Resulting in:
[tex]\[
2 x y^{3/2} + 2 x y^{3/2} + 2 x^{3/2} y^{2} + x^2 y^{5/2} + 4 x^{7/2} y^{5} + 2 x^{3/2} y^{3/2}
\][/tex]
[tex]\[
= 4 x y^{3/2} + 4 x^{3/2} y^2 + x^2 y^{5/2} + 4 x^{7/2} y^5 + 2 x^{3/2} y^{3/2}
\][/tex]
So, the final simplified result is:
[tex]\[
4 x y^{3/2} + x^2 y^{5/2} + 4 x^{7/2} y^5 + 2 x^{3/2} y^{3/2} + 2 x^{3/2} y^2
\][/tex]
