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If the fourth term in the Binomial expansion of $\left [ \frac{2}{x}+x^{log_{8}x} \right ]^{6}\left ( x> 0 \right )$ is $20\times 8^{7} ,$ then a value of $x$ is :
Option 1)
$8^{3}$
Option 2)
$8^{2}$
Option 3)
$8$
Option 4)
$8^{-2}$

Answers (1)

best_answer

$\left [ \frac{2}{x}+x^{log_{8}x} \right ]^{6}$

$T_{4}=T_{3+1}=20\times 8^{7}$ n = 3

$\Rightarrow ^{6}C_{3}\left ( \frac{2}{x} \right )^{3}\left ( x^{log_{8}x} \right )^{3}=20\times 8^{7}$

$\left ( \frac{2}{x} \right )^{3}\left ( x^{log_{8}x} \right )^{3}=\left ( 2^{3} \right )^{7}$

$\left ( \frac{2}{x} \right )\left ( x^{log_{8}x} \right )=\left ( 2 \right )^{7}$

$\frac{x^{log_{8}x}}{x}=2^{6}=8^{2}$

Take log both side with base $8$

$\left ( Log_{8}x \right )^{2}=2+log_{8}x$

$log_{8}x=2 \; or\; -1$

$x=8^{2}$

Option 1)

$8^{3}$

Option 2)
$8^{2}$
Option 3)

$8$

Option 4)

$8^{-2}$

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