**الف)**
```
x/(x^2 + y^2) - y(x - y)^2 / (x^4 - y^4)
```
اول، توجه کنید که `x^4 - y^4 = (x^2 + y^2)(x^2 - y^2) = (x^2 + y^2)(x + y)(x - y)`
```
x/(x^2 + y^2) - y(x - y)^2 / [(x^2 + y^2)(x^2 - y^2)]
= x - y,^2 / [(x^2 + y^2)(x^2 - y^2)]
= x,x - y - y / [(x^2 + y^2)(x^2 - y^2)]
= [x^3 - xy^2 - yx^2 + 2xy^2 - y^3] / [(x^2 + y^2)(x^2 - y^2)]
= [x^3 - y^3 - x^2y + xy^2] / [(x^2 + y^2)(x^2 - y^2)]
= [(x - y)(x^2 + xy + y^2) - xy(x-y)] / [(x^2 + y^2)(x^2 - y^2)]
= [(x - y)(x^2 + xy + y^2 - xy)] / [(x^2 + y^2)(x^2 - y^2)]
= [(x - y)(x^2 + y^2)] / [(x^2 + y^2)(x - y)(x + y)]
= 1 / (x + y)
```
**ب)**
```
(x + 7)/(ax - bx) + (y + 9)/(by - ay)
= (x + 7)/x + (y + 9)/y
= (x + 7)/x - (y + 9)/y
= [(x + 7)y - (y + 9)x] / xy
= [xy + 7y - xy - 9x] / xy
= (7y - 9x) / xy
```
**ج)**
```
(a^2 - b^2)/(a - b) - (a^3 - b^3)/(a^2 - b^2)
= [(a - b)(a + b)]/(a - b) - [(a - b)(a^2 + ab + b^2)]/[(a - b)(a + b)]
= (a + b) - (a^2 + ab + b^2)/(a + b)
= [(a + b)^2 - (a^2 + ab + b^2)] / (a + b)
= (a^2 + 2ab + b^2 - a^2 - ab - b^2) / (a + b)
= ab / (a + b)
```
**د)**
```
(4 + x^2 - 4x) / (2 + x) - 2 - x
= (x^2 - 4x + 4) / (x + 2) - (x + 2)
= (x - 2)^2 / (x + 2) - (x + 2)
= [(x - 2)^2 - (x + 2)^2] / (x + 2)
= [(x^2 - 4x + 4) - (x^2 + 4x + 4)] / (x + 2)
= (x^2 - 4x + 4 - x^2 - 4x - 4) / (x + 2)
= (-8x) / (x + 2)
```
بنابراین، عبارات ساده شده به این صورت هستند:
* **الف:** `1 / (x + y)`
* **ب:** `(7y - 9x) / xy`
* **ج:** `ab / (a + b)`
* **د:** `(-8x) / (x + 2)`
