If the fourth term in the expansion of [(x + x^ _2x)^7] is 4480, then the value of x where…

Question

If the fourth term in the expansion of $${(x + {x^{{{\log }_2}x}})^7}$$ is 4480, then the value of x where x$$\in$$N is equal to :

Accepted Answer

T4 = $${}^7{C_3}{x^4}{({x^{{{\log }_2}x}})^3} = 4480$$

$$ \Rightarrow 35{x^4}{({x^{{{\log }_2}x}})^3} = 4480$$

$$ \Rightarrow {x^4}{({x^{{{\log }_2}x}})^3} = 128$$

take log w.r.t. base 2 we get,

$$4{\log _2}x + 3{\log _2}({x^{{{\log }_2}x}}) = {\log _2}128$$

Let $${\log _2}x = y$$

$4y + 3{y^2} = 7$

$$ \Rightarrow y = 1,{{ - 7} \over 3}$$

$$ \Rightarrow {\log _2}x = 1,{{ - 7} \over 3}$$

$$ \Rightarrow $$ $x = 2,x = {2^{ - 7/3}}$

Solutions · 1