The term independent of x in expansion of {\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}} is

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Answer: C. 210 {\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} ight)^{10}} = {\left( {{{{{\left( {{x^{1/3}}} ight)}^3} + {{\left( {{1^{1/3}}} ight)}^3}} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{{{\left( {\sqrt x } ight)}^2} - {{\left( 1 ight)}^2}} \over {x - {x^{1/2}}}}} ight)^{10}} = {\left( {{{\left( {{x^{1/3}} + 1} ight)\left( {{x^{2/3}} - {x^{1/3}} + 1} ight)} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{\left( {\sqrt x + 1} ight)\left( {\sqrt x - 1} ight)} \over {\sqrt x \left( {\sqrt x - 1} ight)}}} ight)^{10}} = {\left( {\left( {{x^{1/3}} + 1} ight) - {{\left( {\sqrt x + 1} ight)} \over {\sqrt x }}} ight)^{10}} = {\left( {\left( {{x^{1/3}} + 1} ight) - \left( {1 + {1 \over {\sqrt x }}} ight)} ight)^{10}} = {\left( {{x^{1/3}} - {1 \over {{x^{1/2}}}}} ight)^{10}} [Note: For {\left( {{x^\alpha } \pm {1 \over {{x^\beta }}}} ight)^n} the \left( {r + 1} ight)th term with power m of x is r = {{n\alpha - m} \over {\alpha + \beta }}] Here \alpha = {1 \over 3}, \beta = {1 \over 2} and m = 0 then r = {{10 	imes {1 \over 3} - 0} \over {{1 \over 3} + {1 \over 2}}} = {{10} \over 3} 	imes {6 \over 5} = 4 	herefore T5 is the term independent of x. 	herefore T5 = {}^{10}{C_4} = 210

The term independent of $$x$$ in expansion of $${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$$ is

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