Question Number 198474 by cortano12 last updated on 20/Oct/23 $$\mathrm{16000}\:=\:\frac{\mathrm{x}^{\mathrm{3}} }{\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2}} }\: \\ $$$$\:\mathrm{x}=? \\ $$ Answered by Frix last updated on 20/Oct/23 $${n}=\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}−{x}\right)^{\mathrm{2}}…
Question Number 198439 by mr W last updated on 20/Oct/23 Commented by mr W last updated on 20/Oct/23 $${can}\:{you}\:{find}\:{an}\:{other}\:{example} \\ $$$${for}\:{such}\:{a}\:{number}? \\ $$ Commented by…
Question Number 198435 by Abdullahrussell last updated on 20/Oct/23 Answered by Rasheed.Sindhi last updated on 20/Oct/23 $$\mathrm{2}\:\&\:\mathrm{5}\:{make}\:\mathrm{0}\:\left[\mathrm{2}×\mathrm{5}=\mathrm{10}\right] \\ $$$$\mathcal{T}{here}\:{are}\:{more}\:\mathrm{2}'{s}\:{than}\:\mathrm{5}'{s} \\ $$$$\therefore\:{Number}\:{of}\:{trailing}\:{zeros} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:={Number}\:{of}\:\mathrm{5}'{s}\:{as}\:{a}\:{factor} \\ $$$${Counting}\:\mathrm{5}'{s}:…
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Question Number 198431 by hardmath last updated on 19/Oct/23 $$\mathrm{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R} \\ $$$$\mathrm{f}\:\left(\mathrm{3x}\:−\:\mathrm{1}\right)\:=\:\mathrm{x}\:+\:\mathrm{5} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$ Answered by AST last updated on 20/Oct/23 $${f}\left[\mathrm{3}\left(\frac{{x}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}}\right)−\mathrm{1}\right]=\frac{{x}+\mathrm{1}}{\mathrm{3}}+\mathrm{5}\Rightarrow{f}\left({x}\right)=\frac{{x}+\mathrm{16}}{\mathrm{3}} \\…
Question Number 198414 by ajfour last updated on 19/Oct/23 $$\frac{\mathrm{3}}{\mathrm{8}}\left(\mathrm{3}{h}−{p}\right)^{\mathrm{2}} +\mathrm{3}{ph}=\left(\mathrm{3}{h}^{\mathrm{2}} −\mathrm{1}\right) \\ $$$${and} \\ $$$$\frac{\left(\mathrm{3}{h}−{p}\right)^{\mathrm{3}} }{\mathrm{16}}+{p}\left(\mathrm{3}{h}^{\mathrm{2}} −\mathrm{1}\right)={h}^{\mathrm{3}} −{h}−{c} \\ $$$${Find}\:\:{p}\:{and}\:{h}\:\:{in}\:{terms}\:{of}\:\mathrm{0}<{c}<\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}}\centerdot \\ $$ Terms of…
Question Number 198367 by universe last updated on 18/Oct/23 $$\:\:\mathrm{if}\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{{x}^{\mathrm{2}} −\left({b}+\mathrm{1}\right){x}+{b}}{{x}^{\mathrm{2}} −\left({a}+\mathrm{1}\right){x}+{a}}\:\:\left({a}\neq{b}\:\&\:{a},{b}\:\in\:\mathbb{R}\:\sim\:\left\{\mathrm{1}\right\}\right) \\ $$$$\:\:\mathrm{can}\:\mathrm{take}\:\mathrm{all}\:\mathrm{values}\:\mathrm{except}\:\mathrm{two}\:\mathrm{values}\:\alpha\:\&\:\beta \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\alpha+\beta\:=\:\mathrm{0}\:\:\mathrm{then}\:\mid\frac{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} −\mathrm{8}}{\mathrm{ab}}\mid\:\:=\:\:?? \\ $$ Commented by Frix last updated…
Question Number 198372 by cortano12 last updated on 18/Oct/23 $$\:\:\:\frac{\mathrm{1}}{\mathrm{1}×\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{4}×\mathrm{8}}\:+\:\frac{\mathrm{1}}{\mathrm{7}×\mathrm{11}}\:+\frac{\mathrm{1}}{\mathrm{10}×\mathrm{14}}\:+\:\ldots=? \\ $$ Commented by Frix last updated on 19/Oct/23 $$\frac{\mathrm{1}}{\mathrm{8}}+\frac{\sqrt{\mathrm{3}}\pi}{\mathrm{36}} \\ $$ Answered by mr…
Question Number 198295 by cortano12 last updated on 17/Oct/23 $$\:\:\:\mathrm{x}^{\mathrm{3}} −\sqrt[{\mathrm{3}}]{\mathrm{81x}−\mathrm{8}}\:=\:\mathrm{2x}^{\mathrm{2}} −\frac{\mathrm{4}}{\mathrm{3}}\mathrm{x}+\mathrm{2}\: \\ $$ Answered by Frix last updated on 17/Oct/23 $${t}=\mathrm{81}{x}−\mathrm{8} \\ $$$${t}^{\mathrm{3}} −\mathrm{138}{t}^{\mathrm{2}}…
Question Number 198311 by York12 last updated on 17/Oct/23 $${Let}\:\left\{{x}_{{r}} \right\}_{{r}=\mathrm{1}} ^{{n}} {be}\:{n}\:{positive}\:{real}\:{numbers}\:{Show}\:{That}: \\ $$$$\frac{{x}_{\mathrm{1}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{{x}_{\mathrm{2}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} }+…+\frac{{x}_{{n}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}}…