Question Number 18969 by chernoaguero@gmail.com last updated on 02/Aug/17
$$\mathrm{If}\:\left(\frac{{a}}{{b}}\right)+\left(\frac{{b}}{{a}}\right)=\mathrm{2},\:\mathrm{then}\:\mathrm{find}\:\left(\frac{{a}}{{b}}\right)^{\mathrm{10}} −\:\left(\frac{{b}}{{a}}\right)^{\mathrm{10}} . \\ $$
Answered by Joel577 last updated on 02/Aug/17
$$\:\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:\mathrm{2}{ab} \\ $$$$\left({a}\:−\:{b}\right)^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:{a}\:−\:{b}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\:=\:{b} \\ $$$$\mathrm{hence} \\ $$$$\left(\frac{{a}}{{b}}\right)^{\mathrm{10}} \:−\:\left(\frac{{b}}{{a}}\right)^{\mathrm{10}} \:=\:\mathrm{1}\:−\:\mathrm{1}\:=\:\mathrm{0} \\ $$
Commented by chernoaguero@gmail.com last updated on 02/Aug/17
$${Thank}\:{yoj}\:{sir} \\ $$