Question Number 221760 by OmoloyeMichael last updated on 09/Jun/25 Answered by shunmisaki007 last updated on 09/Jun/25 $$\left(\mathrm{For}\:{x}>\mathrm{0}\:\mathrm{and}\:{x}\neq\mathrm{1}.\right) \\ $$$$\mathrm{log}_{{x}} \left(\frac{\mathrm{log}_{\mathrm{4}} \left({x}\right)}{\mathrm{log}_{\mathrm{4}} \left({x}\right)−\mathrm{3}}\right)^{\mathrm{log}_{\mathrm{3}} \left({x}\right)} =\mathrm{2} \\…
Question Number 221697 by fantastic last updated on 09/Jun/25 $${Is}\:\sqrt{{i}}\:{an}\:{imaginary}\:{number}\:\left({i}=\sqrt{−\mathrm{1}}\right)\:{answer}\:{with}\:{logic} \\ $$ Answered by Ghisom last updated on 09/Jun/25 $$\mathrm{i}=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{2}}} \\ $$$${z}={r}\mathrm{e}^{\mathrm{i}\theta} \:\Rightarrow\:\sqrt{{z}}=\sqrt{{r}}\mathrm{e}^{\mathrm{i}\frac{\theta}{\mathrm{2}}} \\ $$$$\Rightarrow\:\sqrt{\mathrm{i}}=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}}…
Question Number 221592 by fantastic last updated on 08/Jun/25 Answered by mr W last updated on 08/Jun/25 Commented by mr W last updated on 08/Jun/25…
Question Number 221626 by fantastic last updated on 08/Jun/25 Answered by mr W last updated on 08/Jun/25 $${side}\:{length}\:{of}\:{square}\:=\mathrm{1} \\ $$$${shaded}\:{area}\:=\frac{\pi×\mathrm{1}^{\mathrm{2}} }{\mathrm{8}}+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} −\frac{\pi}{\mathrm{4}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{4}} \\…
Question Number 221620 by fantastic last updated on 08/Jun/25 $${Solve}\:{for}\:{x} \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{7}{x}}=\sqrt{{x}}\left[{x}\neq\mathrm{0}\right] \\ $$ Answered by fantastic last updated on 08/Jun/25 $${or}\:\left(\mathrm{7}{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} ={x}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$${or}\:\sqrt[{\mathrm{3}}]{\mathrm{7}}.{x}^{\frac{\mathrm{1}}{\mathrm{3}}}…
Question Number 221588 by Nicholas666 last updated on 08/Jun/25 $$ \\ $$$$\:\:\:\:\int\:\frac{\mathrm{8}{t}\:−\:\mathrm{8}{t}^{\:\mathrm{3}} }{{t}^{\:\mathrm{6}} \:+\:\mathrm{6}{t}^{\mathrm{5}} \:+\:\mathrm{3}{t}^{\:\mathrm{4}} \:−\:\mathrm{20}{t}^{\mathrm{3}} \:+\:\mathrm{3}{t}^{\mathrm{2}} \:+\:\mathrm{6}{t}\:+\:\mathrm{1}}\:{dt}\:\:\:\: \\ $$$$ \\ $$ Answered by Frix…
Question Number 221585 by mr W last updated on 08/Jun/25 $${solve}\:{for}\:{x}\:\in{R} \\ $$$$\left({x}^{\mathrm{3}} −\mathrm{6}\right)^{\mathrm{3}} ={x}+\mathrm{6} \\ $$ Commented by mr W last updated on 08/Jun/25…
Question Number 221618 by fantastic last updated on 08/Jun/25 $${solve}\:{for}\:{x} \\ $$$$\mathrm{2}^{{x}} +\mathrm{4}^{{x}} =\mathrm{8}^{{x}} \\ $$ Answered by MathematicalUser2357 last updated on 09/Jun/25 $$\mathrm{log}_{\mathrm{2}} \phi…
Question Number 221586 by Nicholas666 last updated on 08/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{1}\:+\:\mathrm{3}{x}\:}\:{dx} \\ $$$$ \\ $$ Answered by MrGaster last updated on 08/Jun/25 Commented by…
Question Number 221587 by Tawa11 last updated on 08/Jun/25 $$\int_{\:\mathrm{2}} ^{\:\mathrm{3}} \:\frac{\mathrm{tan}^{−\:\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{1}\:\:−\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$ Answered by maths2 last updated on 09/Jun/25 $$=\int_{\mathrm{2}} ^{\mathrm{3}}…