Question

if x+1/x=3*1/3,find the value of x³-1/x³

Answer 1

Dear student
$$\begin{gathered} Wehave, \\ x+\frac{1}{x}=3\frac{1}{3} \\ \Rightarrow x+\frac{1}{x}=\frac{10}{3} \\ Now, \\ x-\frac{1}{x}=\sqrt{{\left(x+\frac{1}{x}\right)}^2-4} \\ \Rightarrow x-\frac{1}{x}=\sqrt{{\left(\frac{10}{3}\right)}^3-4} \\ \Rightarrow x-\frac{1}{x}=\sqrt{\frac{100}{9}-4} \\ \Rightarrow x-\frac{1}{x}=\sqrt{\frac{100-36}{4}} \\ \Rightarrow x-\frac{1}{x}=\sqrt{\frac{64}{4}} \\ \Rightarrow x-\frac{1}{x}=4 \\ Now,x^3-\frac{1}{x^3}={\left(x-\frac{1}{x}\right)}^3+3\left(x\right)\left(\frac{1}{x}\right)\left(x-\frac{1}{x}\right) \\ \Rightarrow x^3-\frac{1}{x^3}=4^3+3\left(4\right) \\ \Rightarrow x^3-\frac{1}{x^3}=76 \end{gathered}$$
Regards