The 9th term of 10(x)5log10⁡x+x⋅x12log10⁡x10 is 450, then the rational value of x is

9 th term of the binomial expansion is 450 and T9=C8 1010(x)5t2x1+12t8 where t=log10x450=4510x5t·x8+4t1=1x5t·x8+4tCompare the powers of x on both sides⇒8+4t−5t=0,(t−2)(5t+2)=0, t=2, −25⇒x=102,10−2/5

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The 9^th term of ${(\frac{\sqrt{10}}{(\sqrt{x})^{5 \log_{10} x}} + x \cdot x^{\frac{1}{2 \log_{10} x}})}^{10}$ is 450, then the rational value of x is

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Detailed Solution

$\begin{array}{rcl} 9 th term of the binomial expansion is 450 \\ and T_{9} & = & {}^{10}C_{8} {(\frac{\sqrt{10}}{(\sqrt{x})^{5 t}})}^{2} {(x^{1 + \frac{1}{2 t}})}^{8} where t = \log_{10} x \\ 450 & = & 45 \frac{10}{x^{5 t}} \cdot x^{8 + \frac{4}{t}} \\ 1 & = & \frac{1}{x^{5 t}} \cdot x^{8 + \frac{4}{t}} \end{array}$

Compare the powers of x on both sides

$\begin{matrix} \Rightarrow 8 + \frac{4}{t} - 5 t = 0, \\ (t - 2) (5 t + 2) = 0, \\ t = 2, - \frac{2}{5} \\ \Rightarrow x = 10^{2}, 10^{- 2 / 5} \end{matrix}$