Category: Algebra

Question Number 218651 by hardmath last updated on 13/Apr/25 Answered by vnm last updated on 14/Apr/25 $$=(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-4\cdot \frac{1}{4})+$$$$(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-4\cdot \frac{1}{8})+$$$$\dots +$$$$(\frac{1}{4n-3}+\frac{1}{4n-2}+\frac{1}{4n-1}+\frac{1}{4n}-4\cdot \frac{1}{4n})=$$$$\underset{{k}=\mathrm{1}}…

Question Number 218603 by hardmath last updated on 12/Apr/25 Commented by mr W last updated on 13/Apr/25 $$seemstrue$$ Commented by mr W last…

Question Number 218482 by Nicholas666 last updated on 10/Apr/25 $$$$$$solveforreal\mathit{x};$$$$$$$$\underset{\:\:\:\mathrm{3}\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{5}} −\mathrm{5}\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{10}\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{10}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{x}}+\mathrm{1}}+\sqrt{\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}}}=\sqrt{\mathrm{3}\:}\:\:\:} {\:} \…

Question Number 218474 by Nicholas666 last updated on 10/Apr/25 $$$$$$solveforreal\mathit{x}$$$$\sqrt{x+\sqrt{x^2+1}}=\sqrt{2}$$$$$$ Answered by mehdee7396 last updated on…

Question Number 218504 by hardmath last updated on 10/Apr/25 $$\mathrm{Let}\mathrm{a}>1\mathrm{fixed}$$$$\mathrm{Solve}\mathrm{for}\mathrm{real}\mathrm{numbers}\mathrm{the}\mathrm{system}$$$$\{\begin{array}{l}{\mathrm{a}}^{\mathbf{x}}+2^{\mathbf{x}}=\mathrm{ay}+2\\ {\mathrm{a}}^{\mathbf{y}}+2^{\mathbf{y}}=\mathrm{az}+2\\ {\mathrm{a}}^{\mathbf{z}}+2^{\mathbf{z}}=\mathrm{ax}+2\end{array}$$ Answered by vnm…