Question Number 5519 by FilupSmith last updated on 17/May/16 $${P}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{{a}^{{i}} }{{b}^{{i}} }\:\:\:\:\:\:\:\:{a}\neq{b},\:{b}\neq\mathrm{0} \\ $$$${P}=?? \\ $$ Commented by Yozzii last updated on 17/May/16…
Question Number 136516 by metamorfose last updated on 22/Mar/21 $${find}\:{all}\:{integers}\:\left({x},{y}\right)\::\:\mathrm{5}^{{x}} =\mathrm{3}^{{x}} +\mathrm{2021}{y} \\ $$ Answered by MJS_new last updated on 23/Mar/21 $$\mathrm{solution}:\:{x}=\mathrm{322}{n}\wedge{y}=\frac{\mathrm{5}^{\mathrm{322}{n}} −\mathrm{3}^{\mathrm{322}{n}} }{\mathrm{2021}}\forall{n}\in\mathbb{N} \\…
Question Number 136500 by rexford last updated on 22/Mar/21 Answered by Dwaipayan Shikari last updated on 22/Mar/21 $${a}−\mathrm{2}{d}+{a}−{d}+{a}+{a}+{d}+{a}+\mathrm{2}{d}+{a}+\mathrm{3}{d}=\mathrm{78} \\ $$$$\Rightarrow\mathrm{6}{a}+\mathrm{3}{d}=\mathrm{78}\Rightarrow\mathrm{2}{a}+{d}=\mathrm{26} \\ $$$${since}\:{a}−\mathrm{2}{d}>\mathrm{1}\Rightarrow{a}>\mathrm{2}{d}+\mathrm{1}\:\:{Greatest}\:{possible}\:{when}\:{a}=\mathrm{2}{d}+\mathrm{1} \\ $$$$\mathrm{2}{a}+{d}=\mathrm{26}\Rightarrow\mathrm{5}{d}+\mathrm{2}=\mathrm{26}\Rightarrow\Rightarrow{d}=\frac{\mathrm{24}}{\mathrm{5}} \\…
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Question Number 70898 by Henri Boucatchou last updated on 09/Oct/19 $$\boldsymbol{{Solve}}\::\: \\ $$$$\left.\mathrm{1}.\right)\:\:\sqrt{\boldsymbol{{x}}−\mathrm{2}}\:+\:\sqrt{\mathrm{4}−\boldsymbol{{x}}}\:=\:\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{6}\boldsymbol{{x}}+\mathrm{11} \\ $$$$\left.\mathrm{2}.\right)\:\:\boldsymbol{{x}}^{\mathrm{4}} \:+\:\boldsymbol{{x}}^{\mathrm{3}} \:−\:\mathrm{2}\boldsymbol{{ax}}^{\mathrm{2}} \:−\:\boldsymbol{{ax}}\:+\:\boldsymbol{{a}}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$ Answered by MJS…
Question Number 5274 by sanusihammed last updated on 03/May/16 $${Please}\:{help}\: \\ $$$$ \\ $$$${The}\:{stock}\:\:{of}\:{umar}\:{pharmacy}\:{in}\:{january}\:{was}\:\mathrm{20240}\:{items}. \\ $$$${its}\:{stock}\:{decline}\:{by}\:\mathrm{20}\:{percent}\:{in}\:{may}\:{due}\:{to}\:{increase}\:{sales}. \\ $$$${What}\:{is}\:{the}\:{amount}\:{of}\:{stock}\:{in}\:{may}\:.\: \\ $$ Commented by Rasheed Soomro last…
Question Number 70780 by MJS last updated on 08/Oct/19 $$\frac{\mathrm{1}}{{t}+\sqrt{{u}}+\sqrt{{v}}}= \\ $$$$=\frac{\left({t}−\sqrt{{u}}+\sqrt{{v}}\right)\left({t}+\sqrt{{u}}−\sqrt{{v}}\right)\left({t}−\sqrt{{u}}−\sqrt{{v}}\right)}{\left({t}+\sqrt{{u}}+\sqrt{{v}}\right)\left({t}−\sqrt{{u}}+\sqrt{{v}}\right)\left({t}+\sqrt{{u}}−\sqrt{{v}}\right)\left({t}−\sqrt{{u}}−\sqrt{{v}}\right)}= \\ $$$$=\frac{{t}^{\mathrm{3}} −\left(\sqrt{{u}}+\sqrt{{v}}\right){t}^{\mathrm{2}} −\left(\sqrt{{u}}−\sqrt{{v}}\right)^{\mathrm{2}} {t}+\left({u}−{v}\right)\left(\sqrt{{u}}−\sqrt{{v}}\right)}{{t}^{\mathrm{4}} −\mathrm{2}\left({u}+{v}\right){t}^{\mathrm{2}} +\left({u}−{v}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\frac{\mathrm{1}}{{t}+\sqrt[{\mathrm{3}}]{{u}}+\sqrt[{\mathrm{3}}]{{v}}}= \\…
Question Number 5226 by sanusihammed last updated on 02/May/16 $${Find}\:{b}\:{in}\:{terms}\:{of}\:{a}\:{if}\:\sqrt{{a}\frac{{a}}{{b}}}\:\:\:=\:\:\left(\frac{{a}}{{b}}\right)^{\frac{\mathrm{1}}{{a}}} \:\:\:\:.\:\:{where}\:{a}\:{and}\:{b}\:{are} \\ $$$${whole}\:{numbers}.\:\: \\ $$ Commented by prakash jain last updated on 02/May/16 $$\left(\frac{{ab}+{a}}{{b}}\right)^{\mathrm{1}/\mathrm{2}} =\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/{a}}…
Question Number 70757 by MJS last updated on 08/Oct/19 $$. \\ $$ Commented by TawaTawa last updated on 07/Oct/19 $$\mathrm{Sir},\:\mathrm{help}\:\mathrm{me}\:\mathrm{with}\:\mathrm{the}\:\mathrm{question}\:\mathrm{number}\:\mathrm{of}\:\mathrm{a}\:\mathrm{question}\:\mathrm{you}\:\mathrm{solved} \\ $$$$\mathrm{sometimes}. \\ $$$$ \\…
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