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Question Number 9119 by tawakalitu last updated on 19/Nov/16
If x = 5^(1/4) + 5^(−1/4) and y = 5^(1/4) − 5^(−1/4) Show that : 5^((x^2 + y^2 )^2 ) = 144
$$\mathrm{If}\:\:\mathrm{x}\:=\:\mathrm{5}^{\mathrm{1}/\mathrm{4}} \:+\:\mathrm{5}^{−\mathrm{1}/\mathrm{4}} \:\:\:\mathrm{and}\:\:\mathrm{y}\:=\:\mathrm{5}^{\mathrm{1}/\mathrm{4}} \:−\:\mathrm{5}^{−\mathrm{1}/\mathrm{4}} \\ $$$$\mathrm{Show}\:\mathrm{that}\::\:\mathrm{5}^{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \right)^{\mathrm{2}} \:} =\:\mathrm{144} \\ $$
Answered by mrW last updated on 19/Nov/16
x^2 =(5^(1/4) +5^(−1/4) )^2 =5^(1/2) +2×5^(1/4) ×5^(−1/4) +5^(−1/2) =5^(1/2) +2+5^(−1/2) y^2 =(5^(1/4) −5^(−1/4) )^2 =5^(1/2) −2×5^(1/4) ×5^(−1/4) +5^(−1/2) =5^(1/2) −2+5^(−1/2) x^2 +y^2 =2×(5^(1/2) +5^(−1/2) ) (x^2 +y^2 )^2 =4×(5+2×5^(1/2) ×5^(−1/2) +5^(−1) )=4×(7+1/5)=((144)/5) 5^((x^2 +y^2 )^2 ) =5^((144)/5) ≈1.35×10^(20) !!! but 5(x^2 +y^2 )^2 =5×((144)/5)=144
$${x}^{\mathrm{2}} =\left(\mathrm{5}^{\mathrm{1}/\mathrm{4}} +\mathrm{5}^{−\mathrm{1}/\mathrm{4}} \right)^{\mathrm{2}} =\mathrm{5}^{\mathrm{1}/\mathrm{2}} +\mathrm{2}×\mathrm{5}^{\mathrm{1}/\mathrm{4}} ×\mathrm{5}^{−\mathrm{1}/\mathrm{4}} +\mathrm{5}^{−\mathrm{1}/\mathrm{2}} =\mathrm{5}^{\mathrm{1}/\mathrm{2}} +\mathrm{2}+\mathrm{5}^{−\mathrm{1}/\mathrm{2}} \\ $$$${y}^{\mathrm{2}} =\left(\mathrm{5}^{\mathrm{1}/\mathrm{4}} −\mathrm{5}^{−\mathrm{1}/\mathrm{4}} \right)^{\mathrm{2}} =\mathrm{5}^{\mathrm{1}/\mathrm{2}} −\mathrm{2}×\mathrm{5}^{\mathrm{1}/\mathrm{4}} ×\mathrm{5}^{−\mathrm{1}/\mathrm{4}} +\mathrm{5}^{−\mathrm{1}/\mathrm{2}} =\mathrm{5}^{\mathrm{1}/\mathrm{2}} −\mathrm{2}+\mathrm{5}^{−\mathrm{1}/\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}×\left(\mathrm{5}^{\mathrm{1}/\mathrm{2}} +\mathrm{5}^{−\mathrm{1}/\mathrm{2}} \right) \\ $$$$\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{4}×\left(\mathrm{5}+\mathrm{2}×\mathrm{5}^{\mathrm{1}/\mathrm{2}} ×\mathrm{5}^{−\mathrm{1}/\mathrm{2}} +\mathrm{5}^{−\mathrm{1}} \right)=\mathrm{4}×\left(\mathrm{7}+\mathrm{1}/\mathrm{5}\right)=\frac{\mathrm{144}}{\mathrm{5}} \\ $$$$\mathrm{5}^{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\mathrm{2}} } =\mathrm{5}^{\frac{\mathrm{144}}{\mathrm{5}}} \approx\mathrm{1}.\mathrm{35}×\mathrm{10}^{\mathrm{20}} \:\:\:\:\:!!! \\ $$$${but} \\ $$$$\mathrm{5}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{5}×\frac{\mathrm{144}}{\mathrm{5}}=\mathrm{144} \\ $$
Commented by mrW last updated on 19/Nov/16
please check the question!
$${please}\:{check}\:{the}\:{question}! \\ $$
Commented by tawakalitu last updated on 19/Nov/16
yes thank you very much. i think it is the question. you are correct. God bless you.
$$\mathrm{yes}\:\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}.\:\mathrm{i}\:\mathrm{think}\:\mathrm{it}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{question}.\:\mathrm{you}\:\mathrm{are}\:\mathrm{correct}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

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