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if-x-2-1-x-2-98-find-x-3-1-x-3-




Question Number 26227 by ktomboy1992 last updated on 23/Dec/17
if x^2 +(1/x^2 )=98 find x^3 +(1/x^3 )
$$\mathrm{if}\:\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }=\mathrm{98}\:\mathrm{find}\:\mathrm{x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} } \\ $$
Answered by $@ty@m last updated on 22/Dec/17
x^2 +(1/x^2 )=(x+(1/x))^2 −2=98  ⇒(x+(1/x))^2 =100  ⇒x+(1/x)=10  (x+(1/x))^3 =x^3 +(1/x^3 )+3(x+(1/x))  ⇒1000=x^3 +(1/x^3 )+30  ⇒x^3 +(1/x^3 )=970
$${x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }=\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} −\mathrm{2}=\mathrm{98} \\ $$$$\Rightarrow\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} =\mathrm{100} \\ $$$$\Rightarrow{x}+\frac{\mathrm{1}}{{x}}=\mathrm{10} \\ $$$$\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{3}} ={x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }+\mathrm{3}\left({x}+\frac{\mathrm{1}}{{x}}\right) \\ $$$$\Rightarrow\mathrm{1000}={x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }+\mathrm{30} \\ $$$$\Rightarrow{x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }=\mathrm{970} \\ $$

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