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Category: Algebra

a-b-c-gt-0-ab-ac-bc-1-Prove-that-a-3-a-b-3-b-c-3-c-2-a-b-c-

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…

n-2-Prove-that-k-1-n-tg-pi-3-1-3-k-3-n-1-k-1-n-ctg-pi-3-1-3-k-3-n-1-

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…

Prove-that-cos20-1-3-cos80-1-3-cos160-1-3-3-9-1-3-6-2-1-3-

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} \\…

Question-217822

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} \\…