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Question Number 207226 by hardmath last updated on 10/May/24

Find: (((2 − (√2))^8 ∙ (6 + 4 (√2))^8 )/((2 + (√2))^8 )) = ?

$$\mathrm{Find}:\:\:\:\frac{\left(\mathrm{2}\:−\:\sqrt{\mathrm{2}}\right)^{\mathrm{8}} \:\centerdot\:\left(\mathrm{6}\:+\:\mathrm{4}\:\sqrt{\mathrm{2}}\right)^{\mathrm{8}} }{\left(\mathrm{2}\:+\:\sqrt{\mathrm{2}}\right)^{\mathrm{8}} }\:=\:? \\ $$

Answered by Sutrisno last updated on 10/May/24

((((2−(√2))(6+4(√2)))/(2+(√2))))^8 ((((2−(√2))(6+4(√2)))/(2+(√2))).((2−(√2))/(2−(√2))))^8 ((((4−4(√2)+2)(6+4(√2)))/(4−2)))^8 ((((6−4(√2))(6+4(√2)))/2))^8 (((36−32)/2))^8 ((4/2))^8 =256

$$\left(\frac{\left(\mathrm{2}−\sqrt{\mathrm{2}}\right)\left(\mathrm{6}+\mathrm{4}\sqrt{\mathrm{2}}\right)}{\mathrm{2}+\sqrt{\mathrm{2}}}\right)^{\mathrm{8}} \\ $$$$\left(\frac{\left(\mathrm{2}−\sqrt{\mathrm{2}}\right)\left(\mathrm{6}+\mathrm{4}\sqrt{\mathrm{2}}\right)}{\mathrm{2}+\sqrt{\mathrm{2}}}.\frac{\mathrm{2}−\sqrt{\mathrm{2}}}{\mathrm{2}−\sqrt{\mathrm{2}}}\right)^{\mathrm{8}} \\ $$$$\left(\frac{\left(\mathrm{4}−\mathrm{4}\sqrt{\mathrm{2}}+\mathrm{2}\right)\left(\mathrm{6}+\mathrm{4}\sqrt{\mathrm{2}}\right)}{\mathrm{4}−\mathrm{2}}\right)^{\mathrm{8}} \\ $$$$\left(\frac{\left(\mathrm{6}−\mathrm{4}\sqrt{\mathrm{2}}\right)\left(\mathrm{6}+\mathrm{4}\sqrt{\mathrm{2}}\right)}{\mathrm{2}}\right)^{\mathrm{8}} \\ $$$$\left(\frac{\mathrm{36}−\mathrm{32}}{\mathrm{2}}\right)^{\mathrm{8}} \\ $$$$\left(\frac{\mathrm{4}}{\mathrm{2}}\right)^{\mathrm{8}} =\mathrm{256} \\ $$

Answered by Rasheed.Sindhi last updated on 10/May/24

((((2 − (√2)) ∙ (6 + 4 (√2)))/((2 + (√2)))))^8 =((((2 − (√2)) ∙ ( (2+(√2) )^2 )/((2 + (√2)))))^8 =( (2−(√2) )(2+(√2) ) )^8 =(4−2)^8 =256

$$\:\left(\frac{\left(\mathrm{2}\:−\:\sqrt{\mathrm{2}}\right)\:\centerdot\:\left(\mathrm{6}\:+\:\mathrm{4}\:\sqrt{\mathrm{2}}\right)}{\left(\mathrm{2}\:+\:\sqrt{\mathrm{2}}\right)}\right)^{\mathrm{8}} \: \\ $$$$\:=\left(\frac{\left(\mathrm{2}\:−\:\sqrt{\mathrm{2}}\right)\:\centerdot\:\left(\:\left(\mathrm{2}+\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} \right.}{\left(\mathrm{2}\:+\:\sqrt{\mathrm{2}}\right)}\right)^{\mathrm{8}} \: \\ $$$$=\left(\:\left(\mathrm{2}−\sqrt{\mathrm{2}}\:\right)\left(\mathrm{2}+\sqrt{\mathrm{2}}\:\right)\:\right)^{\mathrm{8}} \\ $$$$=\left(\mathrm{4}−\mathrm{2}\right)^{\mathrm{8}} =\mathrm{256} \\ $$

Commented by hardmath last updated on 10/May/24

thank you professors

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{professors} \\ $$

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