MODERATE EXERCISES - Symbol Manipulation

(1) Simplify.

  (a)     2h 
        -------
        (h + 1)
     -------------
          (h)

 (b) x2(x + sqrt(7))    Where "sqrt" means the square root of
     ---------------
       2x(x4 + x3)

 (c) x(x - 3) (3 + 5x)
             ------------
             (4x(2x - 6))

 (d) (1/1+z) - 1
     ------------
          z

 (a)      2h            2h     (1)         2h            2
        -------    =  ------  -----  =  --------   =  -------
        (h + 1)       (h + 1)  (h)      (h + 1)(h)    (h + 1)
     -------------
          (h)

 (b) x2(x + sqrt(7))      x2(x + sqrt(7))  
     ---------------  =  ---------------
       2x(x4 + x3)         2x(x3)(x + 1)
  
     =  x2(x + sqrt(7))      (x + sqrt(7))
       ----------------  =  ---------------
          2x4(x + 1)           2x2 (x + 1)

 (c) x(x - 3) (3 + 5x)          x(x - 3) (3 + 5x)        x(x - 3) (3 + 5x)
              -----------   =  -------------------  =  --------------------
             (4x(2x - 6))        (4x)(2x - 6)            (4)(x)(2)(x - 3)

     = (3 + 5x)
       ---------
           8
 
 (d)  (1/1+z) - 1        1     -  (1 + z)         1 - (1 + z)        1 - (1 + z)   (Recall the HINT!)
     ------------  =  -------     -------   =     -----------     = -------------  
          z           (1 + z)     (1 + z)          (1 + z)            z(1 + z)
                    -----------------------    -----------------
                               z                       z
     
     = 1 - 1 - z        z            1
      -----------  =  -------  =  --------
       z(1 + z)       z(1 + z)    ( 1 + z)