Question Number 211262 by mathlove last updated on 03/Sep/24 Answered by mahdipoor last updated on 03/Sep/24 $${f}\left({m}\right)+{f}\left(\mathrm{1}−{m}\right)=\frac{{a}^{{m}−\mathrm{1}} \:}{{a}^{{m}} +{b}}+\frac{{a}^{−{m}} }{{a}^{\mathrm{1}−{m}} +{b}}= \\ $$$$\frac{{a}^{{m}−\mathrm{1}} \left({a}^{\mathrm{1}−{m}} +{b}\right)+{a}^{−{m}}…
Question Number 211255 by ajfour last updated on 02/Sep/24 $$\sqrt{{a}+\sqrt{{b}−{x}}+\sqrt{{b}−\sqrt{{a}+{x}}}}=\mathrm{2}{x} \\ $$$${solve}\:{for}\:{x}.\:\:\:\: \\ $$ Commented by Ghisom last updated on 03/Sep/24 $$\sqrt{{a}+\sqrt{{b}−{x}}+\sqrt{{b}−\sqrt{{a}+{x}}}}=\mathrm{2}{x} \\ $$$${x}=\left({b}−\left(\mathrm{4}{x}^{\mathrm{2}} −\sqrt{{b}−{x}}−{a}\right)^{\mathrm{2}}…
Question Number 211252 by RojaTaniya last updated on 02/Sep/24 Answered by Frix last updated on 02/Sep/24 $$\left[\mathrm{1}\right]×\left({x}−{y}\right)\:\Rightarrow\:{x}^{\mathrm{3}} −{y}^{\mathrm{3}} =\mathrm{39}\left({x}−{y}\right)\:\:\left[\mathrm{1}{a}\right] \\ $$$$\left[\mathrm{2}\right]×\left({y}−{z}\right)\:\Rightarrow\:{y}^{\mathrm{3}} −{z}^{\mathrm{3}} =\mathrm{49}\left({y}−{z}\right)\:\:\left[\mathrm{2}{a}\right] \\ $$$$\left[\mathrm{3}\right]×\left({z}−{x}\right)\:\Rightarrow\:{z}^{\mathrm{3}}…
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Question Number 211250 by RojaTaniya last updated on 01/Sep/24 $${Find}\:{the}\:{number}\:{of}\:\mathrm{4}\:{digit}\:{numbers} \\ $$$$\:{so}\:{that}\:{when}\:{decomposed}\:{into}\:{prime} \\ $$$$\:{factors},\:{have}\:{the}\:{sum}\:{of}\:{prime}\:{factors} \\ $$$$\:{equal}\:{to}\:{the}\:{sum}\:{of}\:{the}\:{exponents}? \\ $$ Answered by mr W last updated on…
Question Number 211251 by hardmath last updated on 01/Sep/24 Commented by a.lgnaoui last updated on 04/Sep/24 Commented by a.lgnaoui last updated on 05/Sep/24 $$\frac{\mathrm{sin}\:\mathrm{C}}{\mathrm{MN}}=\frac{\mathrm{sin}\:\mathrm{M}}{\mathrm{BC}−\mathrm{AB}}=\frac{\mathrm{sin}\:\mathrm{N}}{\mathrm{CD}−\mathrm{AD}}\left(\mathrm{1}\right) \\…
Question Number 211207 by Etimbuk last updated on 31/Aug/24 Answered by mr W last updated on 03/Sep/24 $${e}_{\mathrm{1}} ={a}+{b}+{c}+{d} \\ $$$${e}_{\mathrm{2}} ={ab}+{bc}+{cd}+{da}+{ac}+{bd} \\ $$$${e}_{\mathrm{3}} ={abc}+{bcd}+{cda}+{dab}…
Question Number 211222 by RojaTaniya last updated on 31/Aug/24 $${How}\:{many}\:{integers}\:{are}\:{there} \\ $$$$\:\:\:{such}\:{that},\:\:\mathrm{0}\leqslant{n}\leqslant\mathrm{720}\:{and}\: \\ $$$$\:\:\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\right)\mathrm{720}? \\ $$ Answered by mahdipoor last updated on 31/Aug/24 $$\Rightarrow{n}^{\mathrm{2}}…
Question Number 211201 by efronzo1 last updated on 31/Aug/24 $$\:\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{1}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{2}}\:−\sqrt[{\mathrm{3}}]{\mathrm{2x}−\mathrm{3}}\:=\:\mathrm{0} \\ $$$$\:\:\:\mathrm{x}=? \\ $$ Answered by Rasheed.Sindhi last updated on 31/Aug/24 $$\:\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{1}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{2}}\:−\sqrt[{\mathrm{3}}]{\mathrm{2x}−\mathrm{3}}\:=\:\mathrm{0} \\ $$$$\because−\sqrt[{\mathrm{3}}]{\mathrm{2x}−\mathrm{3}}\:=\sqrt[{\mathrm{3}}]{\mathrm{3}−\mathrm{2x}} \\…
Question Number 211191 by Tawa11 last updated on 30/Aug/24 Answered by mm1342 last updated on 30/Aug/24 $$\left(\left(\mathrm{2}^{{x}} \right)^{{y}} \right)^{{x}} =\left(\mathrm{3}^{{y}} \right)^{{z}} =\mathrm{7}^{{z}} =\mathrm{11}\:\checkmark \\ $$$$…