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Question Number 18969 by chernoaguero@gmail.com last updated on 02/Aug/17

If ((a/b))+((b/a))=2, then find ((a/b))^(10) − ((b/a))^(10) .

$$\mathrm{If}\:\left(\frac{{a}}{{b}}\right)+\left(\frac{{b}}{{a}}\right)=\mathrm{2},\:\mathrm{then}\:\mathrm{find}\:\left(\frac{{a}}{{b}}\right)^{\mathrm{10}} −\:\left(\frac{{b}}{{a}}\right)^{\mathrm{10}} . \\ $$

Answered by Joel577 last updated on 02/Aug/17

  a^2  + b^2  = 2ab  (a − b)^2  = 0        a − b = 0                a = b  hence  ((a/b))^(10)  − ((b/a))^(10)  = 1 − 1 = 0

$$\:\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:\mathrm{2}{ab} \\ $$$$\left({a}\:−\:{b}\right)^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:{a}\:−\:{b}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\:=\:{b} \\ $$$$\mathrm{hence} \\ $$$$\left(\frac{{a}}{{b}}\right)^{\mathrm{10}} \:−\:\left(\frac{{b}}{{a}}\right)^{\mathrm{10}} \:=\:\mathrm{1}\:−\:\mathrm{1}\:=\:\mathrm{0} \\ $$

Commented by chernoaguero@gmail.com last updated on 02/Aug/17

Thank yoj sir

$${Thank}\:{yoj}\:{sir} \\ $$

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