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

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Solution

$(a + b)^{n} \Rightarrow$ General term $\Rightarrow T_{r} =^{n} C_{r} a^{n - r} b^{r}$
$∴ 8^{t h}$ term in $(x^{3 / 2} y^{1 / 2} - x^{1 / 2} y^{3 / 2})^{10}$
$T_{8} \Rightarrow^{10} c_{7} (x^{3 / 2} y^{1 / 2})^{1 / 2} (- 1) (x^{1 / 2} y^{3 / 2})^{7}$
$= \frac{10 \times 8 \times 9 (- 1)}{3 \times 2} (x^{1 / 2} y^{3 / 2}) (x^{3 / 2} y^{2 / 2})$
$= - 10 \times 4 \times 3 (x^{9 / 2 + 7 / 2}) (y^{3 / 2 + 4 / 2})$
$= - 120 (x^{16 / 2} y^{24 / 2})$
$T_{8} = - 120 x^{8} y^{12}$

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Find the 8th term in the expansion of (x3/2y1/2−x1/2y3/2)10

(a+b)n⇒ General term ⇒Tr=nCran−rbr∴8th term in (x3/2y1/2−x1/2y3/2)10T8⇒10c7(x3/2y1/2)1/2(−1)(x1/2y3/2)7=10×8×9(−1)3×2(x1/2y3/2)(x3/2y2/2)=−10×4×3(x9/2+7/2)(y3/2+4/2)=−120(x16/2y24/2)T8=−120x8y12

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