ক. `(a - b)/a + (a + b)/b`

         `= (b(a - b) + a(a + b))/(ab)`

        `= (ab - b^2 + a^2 + ab)/(ab)`

        `= (a^2 + 2ab - b^2)/(ab)`

    উত্তর: `(a^2 + 2ab - b^2)/(ab)`



   খ. `a/(bc) + b/(ca) + c/(ab)`

          `= (a^2 + b^2 + c^2)/(abc)`

    উত্তর: `= (a^2 + b^2 + c^2)/(abc)`

   গ. `(x - y)/x + (y - z)/y + (z - x)/z`

           `= (yz(x - y) + zx(y - z) + xy(z - x))/(xyz)`

           `= (xyz - y^2z + xyz - z^2x + xyz - x^2y)/(xyz)`

           `= (3xyz - x^2y - y^2z - z^2x)/(xyz)`

   উত্তর:  `= (3xyz - x^2y - y^2z - z^2x)/(xyz)`

   ঘ.  `(x + y)/(x - y) + (x - y)/(x + y)`

         `= ((x + y)^2 + (x - y)^2)/((x - y)(x + y))`

        `= (x^2 + 2xy + y^2 + x^2 - 2xy + y^2)/((x + y)(x - y))`

         `= (2x^2 + 2y^2)/(x^2 - y^2)`

         `= (2(x^2 + y^2))/(x^2 - y^2)`

   উত্তর: `= (2(x^2 + y^2))/(x^2 - y^2)`

`1/(x^2 - 3x + 2) + 1/(x^2 - 4x + 3) + 1/(x^2 - 5x + 4)`

 `= 1/(x^2 - 2x - x + 2) + 1/(x^2 - 3x - x + 3) + 1/(x^2 - 4x - x + 4)`

 `= 1/(x(x - 2) - 1(x - 2)) + 1/(x(x - 3) - 1(x - 3)) `

                                 `+ 1/(x(x - 4) - 1(x - 4))`

 `= 1/((x - 2)(x - 1)) + 1/((x - 3)(x - 1)) + 1/((x - 4)(x - 1))`

 `= ((x - 3)(x - 4) + (x - 2)(x - 4) + (x - 2)(x - 3))/((x - 1)(x - 2)(x - 3)(x - 4))`

 `= (x^2 - 7x + 12 + x^2 - 6x + 8 + x^2 - 5x + 6)/((x - 1)(x - 2)(x - 3)(x - 4))`

  উত্তর: `= (3x^2 - 18x + 26)/((x - 1)(x - 2)(x - 3)(x - 4))`

`1/(a^2 - b^2) + 1/(a^2 + ab + b^2) + 1/(a^2 - ab + b^2)`

  `= 1/(a^2 - b^2) + (a^2 - ab + b^2 + a^2 + ab + b^2)/((a^2 + ab + b^2)(a^2 - ab + b^2)`

  `= 1/(a^2 - b^2) + (2a^2 + 2b^2)/((a^2 + b^2 + ab)(a^2 + b^2 - ab))`

  ` = 1/(a^2 - b^2) + (2a^2 + 2b^2)/((a^2 + b^2)^2 - (ab)^2)`

  ` = 1/(a^2 - b^2) + (2a^2 + 2b^2)/(a^4 + 2a^2b^2 + b^4 - a^2b^2)`

   `= 1/(a^2 - b^2) + (2a^2 + 2b^2)/(a^4 + a^2b^2 + b^4)`

   `= (a^4 + a^2b^2 + b^4 + 2(a^2 + b^2)(a^2 - b^2))/((a^2 - b^2)(a^4 + a^2b^2 + b^4))`

    `= (a^4 + a^2b^2 + b^4 + 2(a^4 - b^4))/((a^2 - b^2)(a^4 + a^2b^2 + b^4))`

    `= (a^4 + a^2b^2 + b^4 + 2a^4 - 2b^4)/((a^2 - b^2) {(a^2)^2 + a^2b^2 + (b^2)^2})`

   `= (3a^4 + a^2b^2 - b^4)/((a^2)^3 - (b^2)^3)`

   `= (3a^4 + a^2b^2 - b^4)/(a^6 - b^6)`

   উত্তর: `(3a^4 + a^2b^2 - b^4)/(a^6 - b^6)`

ছ.`1/(x - 2) + 1/(x + 2) + 4/(x^2 - 4)`

  `= (1(x + 2) + 1(x - 2))/((x - 2)(x + 2)) + 4/(x^2 - 4)`

  `= (x + 2 + x - 2)/((x - 2)(x + 2)) + 4/(x^2 - 4)`

  `= (2x)/(x^2 - 4) + 4/(x^2 - 4)`

  `= (2x + 4)/(x^2 - 4)`

  `= (2(x + 2))/((x - 2)(x + 2))`

  `= 2/(x - 2)`

  উত্তর:  `= 2/(x - 2)`


 জ. ` 1/(x^2 - 1) + 1/(x^4 - 1) + 4/(x^8 - 1)`

      `= 1/(x^2 - 1) + 1/(x^2 - 1)(x^2 + 1)) + 4/(x^8 - 1)`

      `= (x^2 + 1 + 1)/((x^2 - 1)(x^2 + 1)) + 4/(x^8 - 1)`

     `= (x^2 + 2)/(x^4 - 1) + 4/((x^4 + 1)(x^4 - 1))`

     `= ((x^2 + 2)(x^4 + 1) + 4)/((x^4 + 1)(x^4 - 1))`

     `= (x^6 + x^2 + 2x^4 + 2 + 4)/((x^4)^2 - 1^2)`

     `= (x^6 + 2x^4 + x^2 + 6)/(x^8 - 1)`

    উত্তর:  `= (x^6 + 2x^4 + x^2 + 6)/(x^8 - 1)`

ক. `a/(x - 3) - a^2/(x^2 - 9)`

     `= a/(x - 3) - a^2/(x^2 - 3^2)`

     `= a/(x - 3) - a^2/((x + 3)(x - 3))`

    ` = (a(x + 3) - a^2)/((x - 3)(x + 3))`

    ` = (ax + 3a - a^2)/((x - 3)(x + 3))`

     `= (ax + 3a - a^2)/(x^2 - 9)`

 উত্তর: `= (ax + 3a - a^2)/(x^2 - 9)`


  খ. `1/(y(x - y)) - 1/(x(x + y))`

     `= (x(x + y) - y(x - y))/(xy(x - y)(x + y))`

     `= (x^2 + xy - xy + y^2)/(xy(x - y)(x + y))`

     `= (x^2 + y^2)/(xy(x - y)(x + y))`

     `= (x^2 + y^2)/(xy(x^2 - y^2)`

      উত্তর: `= (x^2 + y^2)/(xy(x^2 - y^2)`

 গ. `(x + 1)/(1 + x + x^2) - (x - 1)/(1 - x + x^2)`

  `= ((x + 1)(1 - x + x^2) - (x - 1)(1 + x + x^2))/((1 + x + x^2) (1 - x + x^2))`

  `= ((x - x^2 + x^3 + 1 - x + x^2) - (x + x^2 + x^3 - 1 - x - x^2))/((1 + x^2 + x) (1 + x^2 - x))`

 `= ((x^3 + 1) - (x^3 - 1))/((1 + x^2)^2 - x^2)`

 `= ((x^3 + 1) - (x^3 - 1))/(1 + 2x^2 + x^4 - x^2)`

` = (x^3 + 1 - x^3 + 1)/(x^4 + x^2 + 1)`

 `= 2/(x^4 + x^2 + 1)`

   উত্তর :` 2/(x^4 + x^2 + 1)`


 ঘ.  `(a^2 + 16b^2)/(a^2 - 16b^2) - (a - 4b)/(a + 4b)`

 `= (a^2 + 16b^2)/(a^2 - 16b^2) - (a - 4b)/(a + 4b)`

 `= (a^2 + 16b^2)/((a + 4b) (a - 4b)) - (a - 4b)/(a + 4b)`

 `= (a^2 + 16b^2 - (a - 4b)^2)/((a + 4b) (a - 4b))`

 `= (a^2 + 16b^2 - (a^2 - 2.a.4b + 16b^2))/((a + 4b) (a - 4b))`

 `= (a^2 + 16b^2 - (a^2 - 8ab + 16b^2))/((a + 4b) (a - 4b))`

 `= (a^2 + 16b^2 - a^2 + 8ab - 16b^2)/(a^2 - (4b)^2`

 `= (8ab)/(a^2 - 16b^2)`

    উত্তর: `(8ab)/(a^2 - 16b^2)`

   
  ঙ. `1/(x - y) - (x^2 - xy + y^2)/(x^3 + y^3)`

  `= ((x^3 + y^3) - (x - y) (x^2 - xy + y^2))/((x - y) (x^3 + y^3))`

  `= (x^3 + y^3 - (x^3 - x^2y + xy^2 - x^2y + xy^2 - y^3))/((x - y)(x^3 + y^3))`

  `= (x^3 + y^3 - x^3 + x^2y - xy^2 + x^2y - xy^2 + y^3)/((x - y)(x^3 + y^3))`

  `= (2x^2y - 2xy^2 + 2y^3)/((x - y)(x^3 + y^3))`

  `= (2y(x^2 - xy + y^2))/((x - y)(x + y)(x^2 - xy + y^2))`

  `= (2y)/(x^2 - y^2)`

   উত্তর :`(2y)/(x^2 - y^2)`