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Category: Algebra

Question-195194

Question Number 195194 by Abdullahrussell last updated on 26/Jul/23 Answered by Frix last updated on 26/Jul/23 $${a}\:\:\:\:\:{b}\:\:\:\:\:{c}\:\:\:\:\:{d} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{3}\:\:\:\:\mathrm{15}\:\:\:\mathrm{10} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{4}\:\:\:\:\mathrm{12}\:\:\:\:\mathrm{6} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{6}\:\:\:\:\mathrm{12}\:\:\:\:\mathrm{4} \\ $$$$\mathrm{2}\:\:\:\:\mathrm{10}\:\:\:\mathrm{15}\:\:\:\:\mathrm{3}…

Question-195124

Question Number 195124 by Shlock last updated on 25/Jul/23 Answered by witcher3 last updated on 25/Jul/23 $$\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}\leqslant\sqrt{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{y}+\mathrm{1}\right)} \\ $$$$\Leftrightarrow\mathrm{x}+\mathrm{y}+\mathrm{2}\sqrt{\mathrm{xy}}\leqslant\mathrm{xy}+\mathrm{x}+\mathrm{y}+\mathrm{1}\Leftrightarrow\mathrm{xy}+\mathrm{1}\geqslant\mathrm{2}\sqrt{\mathrm{xy}},\mathrm{AM}−\mathrm{GM} \\ $$$$\Rightarrow\forall\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{R}_{+} \sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}\leqslant\sqrt{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{y}+\mathrm{1}\right)} \\ $$$$\Rightarrow\forall\left(\mathrm{a},\mathrm{b}\right)\in\left[\mathrm{1},\infty\left[^{\mathrm{2}} \right.\right.…

Question-195175

Question Number 195175 by valdirmd last updated on 25/Jul/23 Answered by Rasheed.Sindhi last updated on 26/Jul/23 $${x}^{\mathrm{4}} +{x}^{\mathrm{2}} =\mathrm{11}/\mathrm{5}\:,\:\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} =? \\ $$$$\blacktriangleright{Let}\:{y}=\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} \\…

Question-195107

Question Number 195107 by Rupesh123 last updated on 24/Jul/23 Answered by JDamian last updated on 24/Jul/23 $$\mathrm{5}+\mathrm{15}+\mathrm{25}+\:\centerdot\centerdot\centerdot\:+\mathrm{975}+\mathrm{985}+\mathrm{995}= \\ $$$$=\frac{\mathrm{100}}{\mathrm{2}}\left(\mathrm{5}+\mathrm{995}\right)=\mathrm{50000} \\ $$ Terms of Service Privacy…