Question Number 24443 by Tinkutara last updated on 18/Nov/17 $${Solve}\:{for}\:{x}: \\ $$$$\left(\mathrm{10}^{−\mathrm{4}} {x}\right)^{{x}} =\mathrm{4}×\mathrm{10}^{−\mathrm{8}} \\ $$ Answered by ajfour last updated on 18/Nov/17 $$\Rightarrow\:\:{x}\mathrm{ln}\:\left(\mathrm{10}^{−\mathrm{4}} {x}\right)=\mathrm{ln}\:\mathrm{4}−\mathrm{8}…
Question Number 24438 by Tinkutara last updated on 18/Nov/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{to}\:\mathrm{infinite}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{series}\:\frac{{x}}{\mathrm{1}−{x}^{\mathrm{2}} }+\frac{{x}^{\mathrm{2}} }{\mathrm{1}−{x}^{\mathrm{4}} }+\frac{{x}^{\mathrm{4}} }{\mathrm{1}−{x}^{\mathrm{8}} }+…… \\ $$ Commented by prakash jain last updated…
Question Number 89977 by I want to learn more last updated on 20/Apr/20 Commented by jagoll last updated on 20/Apr/20 $$\frac{\mathrm{dx}}{\mathrm{dy}}\:=\:\mathrm{c}.\:\Rightarrow\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{dx}}{\mathrm{dy}} \\ $$$$\mathrm{2x}−\frac{\mathrm{3}}{\mathrm{2}}.\frac{\mathrm{10}}{\mathrm{3}}\sqrt{\mathrm{x}}\:+\mathrm{5}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{c} \\ $$$$\mathrm{2x}−\mathrm{5}\sqrt{\mathrm{x}}+\mathrm{5}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{c}\:=\:\mathrm{0}…
Question Number 155500 by VIDDD last updated on 01/Oct/21 Answered by MJS_new last updated on 01/Oct/21 $${f}\left({x}+{y}\right)−\left({f}\left({x}\right)+{f}\left({y}\right)\right)=\mathrm{0} \\ $$$$\mathrm{2}{axy}−{c}=\mathrm{0} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{only}\:\mathrm{true}\:\forall{x},{y}\in\mathbb{R}\:\mathrm{with}\:{a}=\mathrm{0}\:\Rightarrow\:{c}=\mathrm{0} \\ $$$${a}+{b}+{c}=\mathrm{3}\:\Rightarrow\:{b}=\mathrm{3} \\ $$$${f}\left({x}\right)=\mathrm{3}{x}…
Question Number 155506 by VIDDD last updated on 01/Oct/21 Answered by amin96 last updated on 01/Oct/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{cos}^{\mathrm{2}} \left(\mathrm{1}−{cos}^{\mathrm{2}} \left(\mathrm{1}−\ldots{cos}^{\mathrm{2}} \left(\mathrm{1}−{cos}^{\mathrm{2}} \left({x}\right)\right)\right)\right)}{{sin}\left(\pi×\frac{\left(\sqrt{{x}+\mathrm{4}}−\mathrm{2}\right)\left(\sqrt{{x}+\mathrm{4}}+\mathrm{2}\right)}{\:{x}×\left(\sqrt{{x}+\mathrm{4}}+\mathrm{2}\right)}\right)}= \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{cos}^{\mathrm{2}}…
Question Number 155497 by VIDDD last updated on 01/Oct/21 Answered by amin96 last updated on 01/Oct/21 $$\sqrt{\mathrm{4}−\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\underbrace{\sqrt{\mathrm{4}−\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\ldots}}}=\boldsymbol{\mathrm{x}}\:\: \\ $$$$\sqrt{\mathrm{4}−\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\boldsymbol{\mathrm{x}}}=\boldsymbol{\mathrm{x}}\:\:\:\Rightarrow\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} =\mathrm{4}−\frac{\boldsymbol{\mathrm{x}}}{\mathrm{3}\sqrt{\mathrm{2}}}\:\:\Rightarrow\:\:\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \sqrt{\mathrm{2}}+\boldsymbol{\mathrm{x}}−\mathrm{12}\sqrt{\mathrm{2}}=\mathrm{0} \\ $$$$\boldsymbol{\Delta}=\mathrm{289}\:\:\:\:\:\boldsymbol{\mathrm{x}}=\frac{−\mathrm{1}+\mathrm{17}}{\mathrm{6}\sqrt{\mathrm{2}}}=\frac{\mathrm{8}}{\mathrm{3}\sqrt{\mathrm{2}}} \\ $$$$\boldsymbol{\mathrm{A}}=\mathrm{10}+\boldsymbol{\mathrm{log}}_{\frac{\mathrm{3}}{\mathrm{2}}}…
Question Number 155495 by mathdanisur last updated on 01/Oct/21 $$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c};\mathrm{d}\in\mathbb{R}\:\:\mathrm{verify}\:\:\mathrm{a}+\mathrm{2b}+\mathrm{3c}+\mathrm{4d}=\mathrm{6} \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\boldsymbol{\mathrm{min}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \right) \\ $$ Answered by mr W last updated on…
Question Number 89955 by swizanjere@gmail.com last updated on 20/Apr/20 $$\mathrm{9}^{\mathrm{x}+\mathrm{1}} \nmid\mathrm{28}\left(\mathrm{3}^{\mathrm{x}} \right)+\mathrm{3}=\mathrm{0} \\ $$ Commented by jagoll last updated on 20/Apr/20 $$\mathrm{what}\:\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{notation} \\ $$$$\nmid\:? \\…
Question Number 155481 by mathdanisur last updated on 01/Oct/21 $$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\in\left[\mathrm{1};\infty\right) \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\:\mathrm{a}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}} \:;\:\mathrm{b}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}}} \:;\:\mathrm{c}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{c}}}} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}. \\ $$ Answered by mr W last updated on…
Question Number 155480 by mathdanisur last updated on 01/Oct/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{solution} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{911}\left(\mathrm{xy}\:+\:\mathrm{49}\right) \\ $$ Answered by Rasheed.Sindhi last updated on 01/Oct/21…