Find the 8th term in the expansion of $(x^{3 / 2} y^{1 / 2} - x^{1 / 2} y^{3 / 2})^{10} .$

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Solution

$T_{N} = T_{r + 1} = (- 1)^{r}^{n} C_{r} x^{n - r} y^{r} N = 8,, r = 7, x = x^{3 / 2} y^{1 / 2}, y = x^{1 / 2} y^{3 / 1}, n = 10$

$T_{8} = T_{7 + 1} = (- 1)^{6}^{10} C_{7} (x^{3 / 2} y^{1 / 2})^{3} (x^{1 / 2} y^{3 / 2})^{7}$

$= -^{10} C_{7} x^{9 / 2} \times x^{7 / 2} \times y^{3 / 2} y^{21 / 2}$

$= - 120 x^{8} y^{12}$

$= - 120 x^{8} y^{12}$

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