Category: Algebra

Question Number 154320 by mathdanisur last updated on 17/Sep/21 $$\mathrm{if}\mathrm{a};\mathrm{b};\mathrm{c}>0\mathrm{and}\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=2\mathrm{then}:$$$$\sum _{\mathbf{cyc}}\frac{\mathrm{a}({\mathrm{b}}^4+{\mathrm{c}}^4)}{{\mathrm{b}}^3+{\mathrm{b}}^2\mathrm{c}+{\mathrm{bc}}^2+{\mathrm{c}}^3}⩾1$$ Answered by ghimisi last…

Question Number 23250 by Tinkutara last updated on 28/Oct/17 $$If{(1+x)}^n=C_0+C_{1}x+C_2{x}^2+C_3{x}^3$$$$+\dots +C_n{x}^n,thenprovethat$$$$\underset{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} {\Sigma\Sigma}\left(\frac{{i}}{\:^{{n}} {C}_{{i}} }\:+\:\frac{{j}}{\:^{{n}}…

Question Number 154303 by mathdanisur last updated on 16/Sep/21 $$\mathrm{Solve}\mathrm{in}\mathbb{R}$$$$\frac{{\mathrm{z}}^9-81\mathrm{z}-62}{{\mathrm{z}}^3}=18\sqrt[3]{3\mathrm{z}+2}$$ Commented by MJS_new last updated on 16/Sep/21 $$\mathrm{check}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of} \…

Question Number 154286 by EDWIN88 last updated on 16/Sep/21 $$h=\sqrt[3]{52-47i}+\sqrt[3]{52+47i}$$$$find{h}^2.$$ Commented by MJS_new last updated on 16/Sep/21 $${r}^{\mathrm{1}/\mathrm{3}} \mathrm{e}^{\mathrm{i}\theta/\mathrm{3}} +{r}^{\mathrm{1}/\mathrm{3}}…

Question Number 154275 by mathdanisur last updated on 16/Sep/21 Commented by benhamimed last updated on 16/Sep/21 $$=\Pi\frac{\left(\mathrm{1}+{n}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left({n}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{2}} −\left(\mathrm{2}{n}\right)^{\mathrm{2}} }=\Pi\frac{\left(\mathrm{1}+{n}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left({n}^{\mathrm{2}} −\mathrm{2}{n}+\mathrm{2}\right)\left({n}^{\mathrm{2}}…