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)

\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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