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Author: Tinku Tara

If-2x-a-4b-3-a-3-3a-and-2y-a-4b-3-a-3-3a-then-show-that-x-3-y-3-b-3-

Question Number 202353 by MATHEMATICSAM last updated on 25/Dec/23 $$\mathrm{If}\:\mathrm{2}{x}\:=\:{a}\:+\:\sqrt{\frac{\mathrm{4}{b}^{\mathrm{3}} \:−\:{a}^{\mathrm{3}} }{\mathrm{3}{a}}}\:\mathrm{and} \\ $$$$\mathrm{2}{y}\:=\:{a}\:−\:\sqrt{\frac{\mathrm{4}{b}^{\mathrm{3}} \:−\:{a}^{\mathrm{3}} }{\mathrm{3}{a}}}\:\mathrm{then}\:\mathrm{show}\:\mathrm{that} \\ $$$${x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \:=\:{b}^{\mathrm{3}} \:. \\ $$ Answered by…

Show-that-a-b-b-a-a-b-b-a-1-a-b-a-b-2-ab-

Question Number 202383 by MATHEMATICSAM last updated on 25/Dec/23 $$\mathrm{Show}\:\mathrm{that}\:\frac{{a}\sqrt{{b}}\:−\:{b}\sqrt{{a}}}{{a}\sqrt{{b}}\:+\:{b}\sqrt{{a}}}\:=\:\frac{\mathrm{1}}{{a}\:−\:{b}}\left({a}\:+\:{b}\:−\:\mathrm{2}\sqrt{{ab}}\right). \\ $$ Answered by AST last updated on 25/Dec/23 $$=\frac{\left({a}\sqrt{{b}}−{b}\sqrt{{a}}\right)^{\mathrm{2}} }{{ab}\left({a}−{b}\right)}\:=\frac{{a}^{\mathrm{2}} {b}+{b}^{\mathrm{2}} {a}−\mathrm{2}{ab}\sqrt{{ab}}}{{ab}\left({a}−{b}\right)}=\frac{{a}+{b}−\mathrm{2}\sqrt{{ab}}}{{a}−{b}} \\ $$…

If-A-M-2-2-det-A-0-A-3-A-2-A-Find-the-values-of-det-2A-I-

Question Number 202371 by mnjuly1970 last updated on 25/Dec/23 $$ \\ $$$$\:\:\:{If}\:\:\:\:{A}\:\in\:{M}_{\mathrm{2}×\mathrm{2}} \:,\:{det}\left({A}\:\right)\neq\:\mathrm{0} \\ $$$$\:\:\:\:,\:\:{A}^{\:\mathrm{3}} \:=\:{A}^{\mathrm{2}} \:+{A}\:\Rightarrow\:{Find}\:{the}\: \\ $$$$\:\:\:\:{values}\:{of}\:\:\:{det}\:\left(\mathrm{2}{A}\:−{I}\:\right) \\ $$$$ \\ $$ Answered by…

If-a-1-1-and-a-1-a-2-a-n-n-2-Find-a-2-a-13-

Question Number 202303 by hardmath last updated on 24/Dec/23 $$\mathrm{If}\:\:\:\mathrm{a}_{\mathrm{1}} \:=\:\mathrm{1}\:\:\:\mathrm{and}\:\:\:\mathrm{a}_{\mathrm{1}} \:\centerdot\:\mathrm{a}_{\mathrm{2}} \:\centerdot\:…\:\centerdot\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:=\:\mathrm{n}^{\mathrm{2}} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}_{\mathrm{2}} \:+\:\mathrm{a}_{\mathrm{13}} \:=\:? \\ $$ Answered by MATHEMATICSAM last updated…