Question Number 208071 by hardmath last updated on 03/Jun/24 $$\frac{\mathrm{4}\:\centerdot\:\mathrm{10}^{\boldsymbol{\mathrm{n}}} \:−\:\mathrm{52}}{\mathrm{3}} \\ $$$$\mathrm{The}\:\mathrm{numbers}\:\mathrm{are}\:\:\mathrm{38}\:\:\mathrm{in}\:\mathrm{total} \\ $$$$\mathrm{Find}:\:\:\boldsymbol{\mathrm{n}}\:=\:? \\ $$ Commented by mr W last updated on 04/Jun/24…
Question Number 208020 by Frix last updated on 02/Jun/24 $$\begin{cases}{\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\frac{\mathrm{14}}{\mathrm{625}}}\\{\sqrt{{x}}+\sqrt{{y}}=\mathrm{8}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{solutions}. \\ $$ Answered by efronzo1 last updated on 02/Jun/24 $$\:\:\Rightarrow\mathrm{x}+\mathrm{y}\:+\mathrm{2}\sqrt{\mathrm{xy}}\:=\:\mathrm{64}\: \\ $$$$\:\Rightarrow\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}\:=\:\frac{\mathrm{14}}{\mathrm{625}}\:;\:\mathrm{x}+\mathrm{y}\:=\:\frac{\mathrm{14}}{\mathrm{625}}\mathrm{xy}\: \\…
Question Number 207984 by Thomaseinstein last updated on 02/Jun/24 Answered by Frix last updated on 02/Jun/24 $${p},\:{q}\:>\mathrm{0} \\ $$$$\frac{\mathrm{1}}{{p}^{\mathrm{2}} }−\frac{\mathrm{1}}{{q}^{\mathrm{2}} }=\frac{\mathrm{4}}{\mathrm{3}}\:\Rightarrow\:{q}^{\mathrm{2}} =\frac{\mathrm{3}{p}^{\mathrm{2}} }{\:\mathrm{3}−\mathrm{4}{p}^{\mathrm{2}} } \\…
Question Number 207986 by Thomaseinstein last updated on 02/Jun/24 $$\left({x}−\mathrm{3}\right)^{\sqrt{{x}−\mathrm{3}\:}} \:\:=\:\mathrm{3} \\ $$ Answered by Frix last updated on 02/Jun/24 $${x}−\mathrm{3}>\mathrm{0} \\ $$$$\sqrt{{x}−\mathrm{3}}\:\mathrm{ln}\:\left({x}−\mathrm{3}\right)\:=\mathrm{3} \\ $$$${x}=\mathrm{3}+\mathrm{e}^{\mathrm{2}{t}}…
Question Number 208037 by hardmath last updated on 02/Jun/24 $$\mathrm{Find}:\:\:\:\mathrm{2}\:\mathrm{log}_{\sqrt{\mathrm{5}}} \:\:\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\:\centerdot\:\mathrm{log}_{\sqrt{\boldsymbol{\mathrm{sin}}\:\frac{\boldsymbol{\pi}}{\mathrm{7}}}} \:\:\mathrm{5}\:\:=\:\:? \\ $$ Answered by mr W last updated on 02/Jun/24 $$=\mathrm{2}×\frac{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}{\frac{\mathrm{log}\:\mathrm{5}}{\mathrm{2}}}×\frac{\mathrm{log}\:\mathrm{5}}{\frac{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}{\mathrm{2}}} \\ $$$$=\mathrm{8}×\frac{\cancel{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}}{\cancel{\mathrm{log}\:\mathrm{5}}}×\frac{\cancel{\mathrm{log}\:\mathrm{5}}}{\cancel{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\right)}}…
Question Number 207963 by efronzo1 last updated on 01/Jun/24 $$\:\:\:\underline{\underbrace{\lessdot}} \\ $$ Commented by Frix last updated on 01/Jun/24 $$\mathrm{You}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate}.\:\mathrm{I}\:\mathrm{get} \\ $$$${x}\approx\mathrm{1}.\mathrm{31430946414} \\ $$ Terms…
Question Number 207943 by hardmath last updated on 31/May/24 $$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{2}^{−} } {\mathrm{lim}}\:\frac{\left(\mathrm{x}\:+\:\mathrm{2}\right)\centerdot\left(\mathrm{x}\:+\:\mathrm{1}\right)}{\mid\mathrm{x}\:+\:\mathrm{2}\mid}\:\:=\:\:? \\ $$ Commented by mr W last updated on 31/May/24 $$=\frac{\left(\mathrm{2}+\mathrm{2}\right)\left(\mathrm{2}+\mathrm{1}\right)}{\mid\mathrm{2}+\mathrm{2}\mid}=\mathrm{3}…
Question Number 207930 by hardmath last updated on 30/May/24 $$\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:\:\:\mathrm{number}\:\mathrm{series} \\ $$$$\mathrm{a}_{\boldsymbol{\mathrm{k}}+\mathrm{3}} ^{\mathrm{2}} \:\:+\:\:\mathrm{a}_{\boldsymbol{\mathrm{k}}} \:\:=\:\:\mathrm{a}_{\boldsymbol{\mathrm{k}}+\mathrm{2}} \:\:+\:\:\mathrm{a}_{\boldsymbol{\mathrm{k}}+\mathrm{7}} \\ $$$$\mathrm{find}:\:\:\:\boldsymbol{\mathrm{k}}\:=\:? \\ $$ Commented by mr W…
Question Number 207919 by efronzo1 last updated on 30/May/24 $$\:\:\:\:\downharpoonleft\underline{\:} \\ $$ Commented by Frix last updated on 30/May/24 $${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{0}\vee\mathrm{15} \\ $$…
Question Number 207920 by efronzo1 last updated on 30/May/24 $$\:\:\:\:\mathrm{Given}\:\mathrm{p},\mathrm{q}\:,\mathrm{r}\:\mathrm{and}\:\mathrm{s}\:\mathrm{real}\:\mathrm{positive}\: \\ $$$$\:\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\begin{cases}{\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} =\:\mathrm{r}^{\mathrm{2}} +\mathrm{s}^{\mathrm{2}} }\\{\mathrm{p}^{\mathrm{2}} +\mathrm{s}^{\mathrm{2}} −\mathrm{ps}\:=\:\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} +\mathrm{qr}.}\end{cases} \\ $$$$\:\:\mathrm{Find}\:\:\frac{\mathrm{pq}+\mathrm{rs}}{\mathrm{ps}+\mathrm{qr}}\:. \\…