Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Ask your questions and receive precise answers from experienced professionals across different disciplines. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
### 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]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
