Category: Algebra

Question Number 208076 by hardmath last updated on 04/Jun/24 $${(5-\mid \mathrm{x}\mid )}^{-\frac{1}{3}}({\mathrm{x}}^2-4)<0$$ Answered by TonyCWX08 last updated on 04/Jun/24 $$Thisinequalityaredefinedwhenx\in ⟨-5,5⟩$$$${Two}\:{Possible}\:{Cases} \…

Question Number 208075 by hardmath last updated on 04/Jun/24 $$\mathrm{Find}:\int \sin (2\mathrm{x}-\frac{\pi }{4})=?$$ Answered by TonyCWX08 last updated on 04/Jun/24 $$sin(2x-\frac{\pi }{4})$$$$=sin\left(2x\right)cos\left(\frac{\pi }{4}\right)-sin\left(\frac{\pi }{4}\right)cos\left(2x\right)$$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\left({sin}\left(\mathrm{2}{x}\right)−{cos}\left(\mathrm{2}{x}\right)\right) \…

Question Number 208071 by hardmath last updated on 03/Jun/24 $$\frac{4\cdot {10}^{\mathbf{n}}-52}{3}$$$$\mathrm{The}\mathrm{numbers}\mathrm{are}38\mathrm{in}\mathrm{total}$$$$\mathrm{Find}:\mathbf{n}=?$$ Commented by mr W last updated on 04/Jun/24…

Question Number 207984 by Thomaseinstein last updated on 02/Jun/24 Answered by Frix last updated on 02/Jun/24 $$p,q>0$$$$\frac{\mathrm{1}}{{p}^{\mathrm{2}} }−\frac{\mathrm{1}}{{q}^{\mathrm{2}} }=\frac{\mathrm{4}}{\mathrm{3}}\:\Rightarrow\:{q}^{\mathrm{2}} =\frac{\mathrm{3}{p}^{\mathrm{2}} }{\:\mathrm{3}−\mathrm{4}{p}^{\mathrm{2}} } \…

Question Number 208037 by hardmath last updated on 02/Jun/24 $$\mathrm{Find}:2{\log }_{\sqrt{5}}\sin \frac{\pi }{7}\cdot {\log }_{\sqrt{\mathbf{sin}\frac{\mathit{\pi }}{7}}}5=?$$ Answered by mr W last updated on 02/Jun/24 $$=2\times \frac{\log \left(\sin \frac{\pi }{7}\right)}{\frac{\log 5}{2}}\times \frac{\log 5}{\frac{\log \left(\sin \frac{\pi }{7}\right)}{2}}$$$$=\mathrm{8}×\frac{\cancel{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}}{\cancel{\mathrm{log}\:\mathrm{5}}}×\frac{\cancel{\mathrm{log}\:\mathrm{5}}}{\cancel{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\right)}}…