How to simplify this?

Question

((ax-b)^-1/2) - ((ax-b)^1/2)/b

Answer 1

Ok, so this problem has a negative square root, right?  so that means there is essentially a square root in the denominator because if you move it below the dividing line, you change the sign.  Teachers don't like square roots down there so get rid of that.   multiply the first term by 1 in the form of

 ((ax-b)^1/2)/((ax-b)^1/2).  This removes the radical from the denominator.

 

((ax-b)^-1/2) - ((ax-b)^1/2)/b = 1/((ax-b)^1/2) - ((ax-b)^1/2)/b     [moving the neg. square root down and making it positive]

 

now get rid of it as described above:  ((ax-b)^1/2)/((ax-b)^1/2)* 1/((ax-b)^1/2) - ((ax-b)^1/2)/b 

 

  = (ax-b)^1/2)/(ax-b) - ((ax-b)^1/2)/b.  next create a common denominator and then add them.

 

the common denominator is (b)(ax-b), so times the first term by b/b and the 2nd term by

(ax-b)/(ax-b) to create it:

 

so we have b*(ax-b)^1/2/(b)(ax-b) - ((ax-b)^1/2)(ax-b))/(b)(ax-b)

 

now place that common denom. on the bottom and everything else on top:

 

(b(ax-b)^1/2 - ((ax-b)^1/2)(ax-b)/(b)(ax-b)

 

factor out (ax-b)^1/2 on top from both terms to get:  (ax-b)^1/2(b-(ax-b))/(b)(ax-b)

 

which is reduced to (ax-b)^1/2(2b-ax)/(b(ax-b))  notice (ax-b) in both numerator and denominator.

 

so that pesky radical reduces to ((2b-ax)(ax-b)^-1/2)/b           [note when dividing subtract the powers to

 

get 1/2 - 1 = -1/2 as the power for (ax-b)]

 

your answer is ((2b-ax)(ax-b)^-1/2)/b

 

 

Answer 2

       1             √(ax - b)
------------  −  ------------  =
 √(ax - b)              b 

        b                  ax - b
 -------------  −  --------------  =
  b√(ax - b)        b√(ax - b)

   b - ax + b
----------------  =

  b√(ax - b)

    2b - ax
------------- 
 b√(ax - b)

or

(2b - ax)√(ax - b)
---------------------  =
      b(ax - b)

 (2b - ax)(ax - b)^1/2 
---------------------
      b(ax - b)

Answer 3

(( ax - b)^-1/2 - ((ax -b )^1/2 )/ b =

 

     1 / √ (ax -b)  -(√ (ax -b )/b =

 

      (√(ax -b) */(√(ax -b)* √(ax-b)) - (√ax-b)/b =

     √(ax - b)/ (ax -b) - √ (aX -b)/ b =

     (√(ax -b) (2b-ax)/b(ax-b)

 

 

     [ b(√(ax -b) - ax(√(ax -b)- b√(ax -b)]/ b(ax-b)

          -ax(√(ax -b)) / b(ax -b)

Answer 4

Nisya,

We can re-write the question as

1/(ax - b)^1/2 - ((ax - b)^1/2)/b and then perform subtraction using the LCM of b(ax-b)^1/2. We then get the fraction

(b - ax +b)/ (b(ax-b)^1/2) or (2b - ax) / (b(ax-b)^1/2). Hope this helps.