Réponse :
Bonjour,
Explications étape par étape
[tex]x^2+\dfrac{1}{x^2} =7\\a)\\7^2=49=(x^2+\dfrac{1}{x^2} )^2\\=x^4+\dfrac{1}{x^4}+2*x^2*\dfrac{1}{x^2}\\=x^4+\dfrac{1}{x^4}+2\\\\\boxed{\Longrightarrow\ x^4+\dfrac{1}{x^4}=49-2=47}\\\\b)\\(x+\dfrac{1}{x})^2=x^2+\dfrac{1}{x^2}+2*x*\dfrac{1}{x}=7+2=9\\\\\boxed{x+\dfrac{1}{x}=3}\\[/tex]
c)
[tex]c)\\(x+\dfrac{1}{x})^3=x^3+\dfrac{1}{x^3}+3x^2*\dfrac{1}{x}+3x*\dfrac{1}{x^2}\\=x^3+\dfrac{1}{x^3}+3(x+\dfrac{1}{x})\\\\\boxed{x^3+\dfrac{1}{x^3}=3^3-3*3=18}\\\\d)\\(x^3+\dfrac{1}{x^3})^2=x^6+\dfrac{1}{x^6}+2\\\\\boxed{x^6+\dfrac{1}{x^6}=18^2-2=322}\\[/tex]