Category: Algebra

Question Number 217101 by MathematicalUser2357 last updated on 01/Mar/25 $$\mathrm{is}\mathrm{this}\mathrm{right}\mathrm{when}{(a+bi)}^{c+di}=\mid a+bi{\mid }^{c+di}{e}^{i(c+di)\arg (a+bi)}?$$$$\mathrm{I}\mathrm{had}\mathrm{let}\arg (a+bi)=\{\begin{array}{ll}{\tan }^{-1}\left(\frac{b}{a}\right)& a⩾0\mathrm{and}b⩾0\\ \pi -{\tan }^{-1}(-\frac{b}{a})& a<0\mathrm{and}b⩾0\\ -(\pi -{\tan }^{-1}\left(\frac{b}{a}\right))& a<0\mathrm{and}b<0\\ -{\tan }^{-1}\left(\frac{b}{a}\right)& a⩾0\mathrm{and}b<0\end{array}\mathrm{before}\mathrm{I}\mathrm{solved}\mathrm{it}$$$${(a+bi)}^{c+di}=\mid a+bi{\mid }^{c+di}{e}^{i(c+di)\arg (a+bi)}$$$$=\mid{a}+{bi}\mid^{{c}}…

Question Number 217080 by hardmath last updated on 28/Feb/25 Commented by mr W last updated on 01/Mar/25 $$alsoArea\left(ABCD\right)⩽\sqrt{abcd}$$ Answered by mr W last…

Question Number 217064 by ArshadS last updated on 28/Feb/25 $$\mathrm{Two}\mathrm{numbers}\mathrm{differ}\mathrm{by}6.\mathrm{The}\mathrm{sum}\mathrm{of}\mathrm{their}\mathrm{reciprocals}\mathrm{is}\frac{2}{15}.$$$$\mathcal{D}eterminethenumbers.$$ Answered by A5T last updated on 28/Feb/25 $$\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}=\frac{2}{15}\Rightarrow 15(\mathrm{a}+\mathrm{b})=2\mathrm{ab}$$$$\mathrm{a}−\mathrm{b}=\underset{−} {+}\mathrm{6}\Rightarrow\mathrm{a}=\mathrm{b}\underset{−}…

Question Number 217047 by hardmath last updated on 27/Feb/25 $$\mathrm{Parties}-\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$$$$\mathrm{For}\mathrm{a}\mathrm{convex}\mathrm{quadrilateral}$$$$\mathrm{Prove}\mathrm{that}:$$$$\mathrm{Area}\left(\mathrm{ABCD}\right)⩽\frac{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2+{\mathrm{d}}^2}{4}$$ Terms of Service…

Question Number 216983 by hardmath last updated on 26/Feb/25 Answered by mr W last updated on 26/Feb/25 $$\overrightarrow{\mathit{A}}=(a_1,a_2,..,a_n)$$$$\overset{\rightarrow} {\boldsymbol{{B}}}=\left({b}_{\mathrm{1}}…