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Category: Algebra

Simplify-2-2-2-2-2-2-2-

Question Number 146964 by mathdanisur last updated on 16/Jul/21 $${Simplify}: \\ $$$$\frac{\sqrt{\mathrm{2}}\:\centerdot\:\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{2}}}\:\centerdot\:\sqrt{\mathrm{2}\:-\:\sqrt{\mathrm{2}}}}{\:\sqrt{\mathrm{2}\sqrt{\mathrm{2}}}}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 16/Jul/21 $${x}\:=\:\frac{\sqrt{\mathrm{2}}\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}.\sqrt{\mathrm{2}−\sqrt{\mathrm{2}}}}{\:\sqrt{\mathrm{2}\sqrt{\mathrm{2}}}} \\ $$$${x}\:=\:\frac{\sqrt{\mathrm{2}}\sqrt{\left(\mathrm{2}+\sqrt{\mathrm{2}}\right)\left(\mathrm{2}−\sqrt{\mathrm{2}}\right)}}{\:\sqrt{\mathrm{2}\sqrt{\mathrm{2}}}} \\…

sin-x-cos-x-lt-0-

Question Number 146961 by mathdanisur last updated on 16/Jul/21 $$\sqrt{{sin}\left({x}\right)}\:\centerdot\:{cos}\left({x}\right)\:<\:\mathrm{0} \\ $$ Answered by TheHoneyCat last updated on 16/Jul/21 $$\mathrm{this}\:\mathrm{expression}\:_{''\sqrt{}''} \:\mathrm{requires}\:\mathrm{that}\:{sin}\left({x}\right)>\mathrm{0} \\ $$$$\mathrm{thus}\:{cos}\left({x}\right)<\mathrm{0} \\ $$$$…

Question-146960

Question Number 146960 by mathdanisur last updated on 16/Jul/21 Answered by TheHoneyCat last updated on 16/Jul/21 $$\mathrm{let}\:\mathscr{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{fuction}\:\mathrm{that}\:\mathrm{gives}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{surface} \\ $$$$\mathrm{let}\:\Delta={ABO} \\ $$$$\mathrm{let}\:{x}\:\mathrm{be}\:\mathrm{the}\:\mathrm{hatched}\:\mathrm{area} \\ $$$$\mathrm{let}\:{D}\:\mathrm{be}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{between}\:\left[\mathrm{AO}\right]\:\mathrm{and}\:\mathrm{the}\:\mathrm{circle}\:\mathscr{C}\left({O},{OR}\right) \\ $$$$\mathrm{let}\:{y}\:\mathrm{be}\:\mathrm{the}\:\mathrm{portion}\:{AOD}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}…

If-ax-2-b-x-c-for-all-x-a-b-gt-0-then-prove-minimum-value-of-27ab-2-is-4c-2-

Question Number 15869 by prakash jain last updated on 14/Jun/17 $$\mathrm{If}\:{ax}^{\mathrm{2}} +\frac{{b}}{{x}}\geqslant{c}\:\mathrm{for}\:\mathrm{all}\:{x},{a},{b}>\mathrm{0}\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{27}{ab}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{4}{c}^{\mathrm{2}} . \\ $$ Commented by prakash jain last…

if-arg-i-z-i-pi-4-find-RemZ-ImZ-

Question Number 146923 by mathdanisur last updated on 16/Jul/21 $${if}\:\:\:\:{arg}\:\left(\frac{{i}\:-\:{z}}{{i}}\right)\:=\:\frac{\pi}{\mathrm{4}} \\ $$$${find}\:\:\:\:{RemZ}\:+\:{ImZ}\:=\:? \\ $$ Answered by gsk2684 last updated on 16/Jul/21 $${arg}\left(\mathrm{1}+{iz}\right)=\frac{\pi}{\mathrm{4}} \\ $$$${arg}\left(\mathrm{1}+{i}\left({x}+{iy}\right)\right)=\frac{\pi}{\mathrm{4}} \\…