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given-27-a-64-b-216-c-72-find-2020abc-3ab-3ac-3bc-ab-ac-bc-2020abc-




Question Number 88128 by john santu last updated on 08/Apr/20
given 27^a  = 64^b  = 216^c  = 72  find ((2020abc)/(3ab+3ac+3bc)) + ((ab+ac+bc)/(2020abc))
$${given}\:\mathrm{27}^{{a}} \:=\:\mathrm{64}^{{b}} \:=\:\mathrm{216}^{{c}} \:=\:\mathrm{72} \\ $$$${find}\:\frac{\mathrm{2020}{abc}}{\mathrm{3}{ab}+\mathrm{3}{ac}+\mathrm{3}{bc}}\:+\:\frac{{ab}+{ac}+{bc}}{\mathrm{2020}{abc}} \\ $$
Answered by $@ty@m123 last updated on 08/Apr/20
Given 27^a  = 64^b  = 216^c  = 72  27=72^((1/a) ) ...(i)  64=72^(1/b)  ...(ii)  216=72^(1/c)  ...(iii)  ∴ 27×64×216=72^((1/a)+(1/b)+(1/c))   ⇒3^3 ×2^6 ×(2×3)^3 =(3^2 ×2^3 )^m  {where m=(1/a)+(1/b)+(1/c)  ⇒3^6 ×2^9 =3^(2m) ×2^(3m)   ⇒m=3 ...(iii)  The given expression  ((2020)/3)×(1/m)+(m/(2020))  =((2020)/9)+(3/(2020))  =224((8107)/(18180))
$${Given}\:\mathrm{27}^{{a}} \:=\:\mathrm{64}^{{b}} \:=\:\mathrm{216}^{{c}} \:=\:\mathrm{72} \\ $$$$\mathrm{27}=\mathrm{72}^{\frac{\mathrm{1}}{{a}}\:} …\left({i}\right) \\ $$$$\mathrm{64}=\mathrm{72}^{\frac{\mathrm{1}}{{b}}} \:…\left({ii}\right) \\ $$$$\mathrm{216}=\mathrm{72}^{\frac{\mathrm{1}}{{c}}} \:…\left({iii}\right) \\ $$$$\therefore\:\mathrm{27}×\mathrm{64}×\mathrm{216}=\mathrm{72}^{\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}} \\ $$$$\Rightarrow\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{6}} ×\left(\mathrm{2}×\mathrm{3}\right)^{\mathrm{3}} =\left(\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{3}} \right)^{{m}} \:\left\{{where}\:{m}=\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\right. \\ $$$$\Rightarrow\mathrm{3}^{\mathrm{6}} ×\mathrm{2}^{\mathrm{9}} =\mathrm{3}^{\mathrm{2}{m}} ×\mathrm{2}^{\mathrm{3}{m}} \\ $$$$\Rightarrow{m}=\mathrm{3}\:…\left({iii}\right) \\ $$$${The}\:{given}\:{expression} \\ $$$$\frac{\mathrm{2020}}{\mathrm{3}}×\frac{\mathrm{1}}{{m}}+\frac{{m}}{\mathrm{2020}} \\ $$$$=\frac{\mathrm{2020}}{\mathrm{9}}+\frac{\mathrm{3}}{\mathrm{2020}} \\ $$$$=\mathrm{224}\frac{\mathrm{8107}}{\mathrm{18180}} \\ $$
Commented by john santu last updated on 08/Apr/20
same. but i don′t understand   why the answer (1/3)?
$${same}.\:{but}\:{i}\:{don}'{t}\:{understand}\: \\ $$$${why}\:{the}\:{answer}\:\frac{\mathrm{1}}{\mathrm{3}}? \\ $$

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