First slide

Binomial theorem for positive integral Index

Question

The term independent of $y$ in the expansion of ${(y^{1 / 6} - y^{- 1 / 3})}^{9}$ is

Easy

A

84
B

$8.4$
C

$- 0.84$
D

$- 84$

Solution

Given expansion ${(y^{1 / 6} - y^{- 1 / 3})}^{9}$ is

We have general term in the expansion ${(x + a)}^{n}$

$(∴ T_{r + 1} =^{n} C_{r} x^{n - r} {(a)}^{r}$ be the expansion of ${(x + a)}^{n})$

$T_{r + 1} =^{9} C_{r} {(y^{1 / 6})}^{9 - r} {(- y^{- 1 / 3})}^{r}$

$T_{r + 1} =^{9} C_{r} {(- 1)}^{r} y^{\frac{9 - 3 r}{6}} .............. (1)$

$y^{\frac{9 - 3 r}{6}}$ compare with $y^{0}$ because of independent term

$\Rightarrow y^{\frac{9 - 3 r}{6}} = y^{0}$

And this will contain $y$ if $\frac{9 - 3 r}{6} = 0$

$∵ r = 3$ substitute in Eqn (1)

$T_{3 + 1} =^{9} C_{3} {(- 1)}^{3} y^{\frac{9 - 3 \times 3}{6}}$

$T_{4} = - \frac{9 \times 8 \times 7}{3 \times 2 \times 1} y^{0} = - 84$

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