Answer :
To solve the expression [tex]\(\frac{x}{(x+y)^2}+\frac{y}{x^2-y^2}\)[/tex], we should break it down into its constituent parts and examine each term separately.
### Step-by-Step Solution
1. Understand the Structure of the Expression:
The expression is a sum of two fractions:
[tex]\[ \frac{x}{(x+y)^2} + \frac{y}{x^2 - y^2} \][/tex]
2. Analyze the Denominators:
- For the first term [tex]\(\frac{x}{(x+y)^2}\)[/tex]:
The denominator is [tex]\((x+y)^2\)[/tex].
- For the second term [tex]\(\frac{y}{x^2-y^2}\)[/tex]:
The denominator is [tex]\(x^2 - y^2\)[/tex], which can be factored as [tex]\((x+y)(x-y)\)[/tex] due to the difference of squares.
3. Rewriting the Second Term:
Rewriting the second term with its factored form helps in better understanding:
[tex]\[ \frac{y}{x^2 - y^2} = \frac{y}{(x+y)(x-y)} \][/tex]
4. Combine to Form the Complete Expression:
Bringing the two terms together, we get:
[tex]\[ \frac{x}{(x+y)^2} + \frac{y}{(x+y)(x-y)} \][/tex]
5. Consider the First Fraction In Its Standard Form:
The first term remains:
[tex]\[ \frac{x}{(x+y)^2} \][/tex]
6. Summarize the Entire Expression:
Combining everything, we have our simplified expression intact:
[tex]\[ \frac{x}{(x+y)^2} + \frac{y}{(x+y)(x-y)} \][/tex]
These steps should provide a clear understanding of how the expression is structured and how each part contributes to the whole sum. This thorough examination shows that our final expression to this question is indeed:
[tex]\[ \boxed{\frac{x}{(x+y)^2} + \frac{y}{x^2 - y^2}} \][/tex]
### Step-by-Step Solution
1. Understand the Structure of the Expression:
The expression is a sum of two fractions:
[tex]\[ \frac{x}{(x+y)^2} + \frac{y}{x^2 - y^2} \][/tex]
2. Analyze the Denominators:
- For the first term [tex]\(\frac{x}{(x+y)^2}\)[/tex]:
The denominator is [tex]\((x+y)^2\)[/tex].
- For the second term [tex]\(\frac{y}{x^2-y^2}\)[/tex]:
The denominator is [tex]\(x^2 - y^2\)[/tex], which can be factored as [tex]\((x+y)(x-y)\)[/tex] due to the difference of squares.
3. Rewriting the Second Term:
Rewriting the second term with its factored form helps in better understanding:
[tex]\[ \frac{y}{x^2 - y^2} = \frac{y}{(x+y)(x-y)} \][/tex]
4. Combine to Form the Complete Expression:
Bringing the two terms together, we get:
[tex]\[ \frac{x}{(x+y)^2} + \frac{y}{(x+y)(x-y)} \][/tex]
5. Consider the First Fraction In Its Standard Form:
The first term remains:
[tex]\[ \frac{x}{(x+y)^2} \][/tex]
6. Summarize the Entire Expression:
Combining everything, we have our simplified expression intact:
[tex]\[ \frac{x}{(x+y)^2} + \frac{y}{(x+y)(x-y)} \][/tex]
These steps should provide a clear understanding of how the expression is structured and how each part contributes to the whole sum. This thorough examination shows that our final expression to this question is indeed:
[tex]\[ \boxed{\frac{x}{(x+y)^2} + \frac{y}{x^2 - y^2}} \][/tex]
