If the term independent of x in the expansion of ( x - 1 x^ 2/3 + x^ 1/3 + 1 + x - 1 x - x^ 1/2 )¹⁰ , x > 1 , is 215040 , then the value of the positive real number is:

Step-by-step solution

aligned & x - 1 x^ 2/3 + x^ 1/3 + 1 + x - 1 x - x^ 1/2 & = (x^ 1/3 - 1)(x^ 2/3 + x^ 1/3 + 1) x^ 2/3 + x^ 1/3 + 1 + ( x - 1)( x + 1) x ( x - 1) & = (x^ 1/3 - 1) + x + 1 x & = (x^ 1/3 - 1) + (1 + x^ -1/2 ) & = (x^ 1/3 + x^ -1/2 ) aligned The given expression becomes ( x^ 1/3 + x^ -1/2 )¹⁰ . The general term is given by: T_ r+1 = ¹⁰C_r ( x^ 1/3 )^ 10-r ( x^ -1/2 )^r T_ r+1 = ¹⁰C_r ¹⁰ x^ 10-r 3 - r 2 For the term independent of x , the exponent of x must be zero: 10-r 3 - r 2 = 0 20 - 2r - 3r = 0 5r = 20 r = 4 The term