a)
[tex]\displaystyle x-\frac1x=3\Rightarrow \left(x-\frac1x\right)^2=3^2\Rightarrow x^2-2\cdot x\cdot\frac1x+\frac1{x^2}=9\Rightarrow x^2+\frac{1}{x^2}=11[/tex]
b)
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]
[tex]\displaystyle x^3-\frac{1}{x^3}=x^3-\left(\frac1{x}\right)^3=\left(x-\frac1x\right)\left(x^2+x\cdot\frac1x+\frac1{x^2}\right)=3(11+1)=36[/tex]
c)
[tex]\displaystyle x^2+\frac1{x^2}=11\Rightarrow \left(x^2+\frac1{x^2}\right)^2=11^2\Rightarrow x^4+2\cdot x^2\cdot\frac{1}{x^2}+\frac{1}{x^4}=121\Rightarrow x^4+\frac{1}{x^4}=121-2=119[/tex]
