Question Number 222703 by gabthemathguy25 last updated on 05/Jul/25 Commented by gabthemathguy25 last updated on 06/Jul/25 $${Where}\:{the}\:{circles}\:{radius}\:{is}\:{R}\:=\:\mathrm{7} \\ $$ Answered by mr W last updated…
Question Number 222697 by gabthemathguy25 last updated on 05/Jul/25 Answered by gregori last updated on 05/Jul/25 $$\:\sqrt[{\mathrm{3}}]{{e}^{\mathrm{2ln}\:\left(\mathrm{81}\right)} }\:=\:\left(\mathrm{81}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)=\:\mathrm{9}^{\frac{\mathrm{4}}{\mathrm{3}}} \\ $$$$\:\sqrt[{\mathrm{4}}]{\left(\mathrm{3}^{\mathrm{log}\:_{\mathrm{9}} \left(\mathrm{9}^{\mathrm{2}} \right)} \right)^{\mathrm{2}} }\:=\:\sqrt{\mathrm{9}}=\:\mathrm{9}^{\frac{\mathrm{1}}{\mathrm{2}}}…
Question Number 222711 by sonukgindia last updated on 05/Jul/25 Answered by gabthemathguy25 last updated on 05/Jul/25 $$\left({ABC}\right)_{\mathrm{16}} =\mathrm{10}×\mathrm{16}^{\mathrm{2}} +\mathrm{11}×\mathrm{16}^{\mathrm{1}} +\mathrm{12}×\mathrm{16}^{\mathrm{0}} \\ $$$$\Rightarrow\mathrm{10}×\mathrm{256}+\mathrm{11}×\mathrm{16}+\mathrm{12}=\mathrm{2560}+\mathrm{176}+\mathrm{12}=\mathrm{2748} \\ $$$$\mathrm{its}\:\mathrm{1},\:\mathrm{2748}. \\…
Question Number 222705 by MrGaster last updated on 05/Jul/25 $$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}{x}\mathrm{arctan}{x}}{{x}}{dx} \\ $$ Answered by gabthemathguy25 last updated on 05/Jul/25 $${this}\:{is}\:{non}\:{elementary}\:{however}\:{i}\:{can}\:{still}\:{evaluate}\:{this} \\ $$$$\mathrm{tan}^{−\mathrm{1}} =\underset{{n}\:=\:\mathrm{0}}…
Question Number 222662 by MrGaster last updated on 04/Jul/25 Commented by MrGaster last updated on 04/Jul/25 $$\exists\bigtriangleup{ABC},\angle{B}=\mathrm{90}°,{BA}={AC},{BD}=\mathrm{3}\wedge\angle{CED}=\mathrm{45}°,{CE}=\sqrt{\mathrm{2}}{AE},{CE}=? \\ $$ Commented by mr W last updated…
Question Number 222679 by fantastic last updated on 04/Jul/25 $${If}\:{x}=\underset{{x}=\mathrm{1}} {\overset{\mathrm{10}} {\prod}}{x}\:{then}\:\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{2}} {x}}+\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{3}} {x}}+\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{4}} {x}}…+\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{10}} {x}}=?? \\ $$ Answered by Tawa11 last updated on 04/Jul/25…
Question Number 222673 by ajfour last updated on 04/Jul/25 Commented by ajfour last updated on 04/Jul/25 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Find}\:{small}\:{circle}\:{radius}\:\boldsymbol{{a}}. \\ $$$${Oh}!\:{i}\:{forgot}\:{to}\:{say}\:{almost}-\:{ABCD}\:{is}\:{sq}. \\ $$ Answered by mr W…
Question Number 222652 by Mingma last updated on 03/Jul/25 Commented by Ghisom last updated on 03/Jul/25 $$\left[\left({A}−{B}\right)\cup\left({B}−{C}\right)\cup\left({C}−{A}\right)\right]'= \\ $$$$=\left[\left({A}\cup{B}\cup{C}\right)−\left({A}\cap{B}\cap{C}\right)\right]'= \\ $$$$={A}\cap{B}\cap{C} \\ $$ Terms of…
Question Number 222638 by Mingma last updated on 03/Jul/25 Answered by gabthemathguy25 last updated on 03/Jul/25 $$\mathrm{cot}\:{A}\:+\:\mathrm{cot}\:{B}\:+\:\mathrm{cot}\:{C}\:=\:\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }{\mathrm{4}{S}} \\ $$$${where}\:{a},{b},{c}\:=\:{side}\:{lengths},\:{S}={area}\:{of}\:{triangle} \\ $$$${S}={r}\centerdot{s}\:\Rightarrow\:\frac{\mathrm{1}}{{S}}=\frac{\mathrm{1}}{{rs}} \\…
Question Number 222639 by Mingma last updated on 03/Jul/25 Answered by gabthemathguy25 last updated on 03/Jul/25 $$\frac{\mathrm{1}}{\mathrm{2}}\mathrm{8}\:{cm}=\mathrm{4}\:{cm} \\ $$$$\mathrm{Slant}\:\mathrm{height}\:=\:\sqrt{\mathrm{10}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} }=\sqrt{\mathrm{100}+\mathrm{16}}=\sqrt{\mathrm{116}}\approx\mathrm{10}.\mathrm{77}\:\mathrm{cm} \\ $$$$\mathrm{cos}\left(\theta\right)=\frac{\mathrm{10}.\mathrm{77}^{\mathrm{2}} +\mathrm{10}.\mathrm{77}^{\mathrm{2}} −\mathrm{8}^{\mathrm{2}}…