If the fourth term in the Binomial expansion of \left(\left(\right. \frac{2}{x} + x^{\left(log\right)_{8} x} \left.\right)\right)^{6}(x

Engineering

Mathematics

Introduction to Binomial Theorem

Question

If the fourth term in the Binomial expansion of ${(\frac{2}{x} + x^{\log_{8} x})}^{6}$ (x > 0) is 20 × 8⁷, then a value of x is

8³

8²

8^–2

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$T_{4} = T_{3 + 1} = (\frac{6}{3}) {(\frac{2}{x})}^{3} . {(x^{\log_{8} x})}^{3}$

$20 \times 8^{7} = \frac{160}{x^{3}} . x^{3 \log_{8} x}$

$8^{6} = x^{\log_{2} x} - 3$

$2^{18} = x^{\log_{2} x - 3}$

⇒ 18 = (og₂ x – 3) (log₂ x)

Let log₂ x = t

⇒ t₂ – 3t – 18 = 0

⇒ (t – 6)(t + 3) = 0

⇒ t = 6, – 3

Log₂ x = 6 ⇒ x = 2⁶ = 8²

Log₂ x = –3 ⇒ x = 2^–3 = 8^–1