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Question Number 217872 by hardmath last updated on 22/Mar/25 $$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:>\:\mathrm{0} \\ $$$$\mathrm{ab}\:+\:\mathrm{ac}\:+\:\mathrm{bc}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt{\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{a}}\:\:+\:\:\sqrt{\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{b}}\:\:+\:\:\sqrt{\mathrm{c}^{\mathrm{3}} \:+\:\mathrm{c}}\:\:\geqslant\:\mathrm{2}\:\sqrt{\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}} \\ $$ Terms of Service Privacy…
Question Number 217843 by hardmath last updated on 22/Mar/25 Commented by AntonCWX8 last updated on 22/Mar/25 $${What}\:{does}\:{the}\:{R}\:{and}\:{r}\:{represents}? \\ $$ Commented by hardmath last updated on…
Question Number 217871 by hardmath last updated on 22/Mar/25 $$\mathrm{n}\:\geqslant\:\mathrm{2} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\:\mathrm{tg}\:\left[\:\frac{\pi}{\mathrm{3}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{3}^{\boldsymbol{\mathrm{n}}} \:−\:\mathrm{1}}\right)\right]\:=\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\:\mathrm{ctg}\:\left[\:\frac{\pi}{\mathrm{3}}\:\left(\mathrm{1}\:−\:\frac{\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{3}^{\boldsymbol{\mathrm{n}}} \:−\:\mathrm{1}}\right)\right] \\ $$ Answered…
Question Number 217857 by hardmath last updated on 22/Mar/25 $$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos20}°}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{cos80}°}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{cos160}°}\:=\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{3}\:\centerdot\:\sqrt[{\mathrm{3}}]{\mathrm{9}}\:−\:\mathrm{6}}{\mathrm{2}}} \\ $$ Answered by Frix last updated on 22/Mar/25 $$\mathrm{Staying}\:\mathrm{in}\:\mathbb{R}\:\Rightarrow\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{20}°}=−\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{160}°}\:\Rightarrow \\ $$$$\mathrm{cos}\:\mathrm{80}°\:=\frac{\mathrm{3}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}}−\mathrm{3}\approx.\mathrm{12} \\…
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Question Number 217789 by hardmath last updated on 21/Mar/25 Commented by mr W last updated on 21/Mar/25 $$?=\sqrt{\left(\mathrm{27}+\mathrm{12}+\mathrm{15}\right)^{\mathrm{2}} −\mathrm{4}×\mathrm{27}×\mathrm{15}}=\mathrm{36} \\ $$ Commented by mr W…
Question Number 217822 by Ari last updated on 21/Mar/25 Answered by mr W last updated on 22/Mar/25 $$\left.{a}\right) \\ $$$$\mathrm{0}.\mathrm{10}×\mathrm{0}.\mathrm{05}=\mathrm{0}.\mathrm{5\%} \\ $$$$\left.{b}\right) \\ $$$${can}\:{not}\:{be}\:{determined},\:{but}\:{at}\:{least} \\…
Question Number 217802 by Ismoiljon_008 last updated on 21/Mar/25 Commented by Ismoiljon_008 last updated on 21/Mar/25 $$\:\:\:{a}+{b}\:=\:{c}\:\:\:{Then}\:\:{R}\:=\:? \\ $$$$\:\:\:{Help}\:{me},\:{please}\: \\ $$$$ \\ $$ Commented by…