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The-value-of-18-3-7-3-3-18-7-25-3-6-6-243-2-15-181-4-20-27-8-15-9-16-6-3-32-64-is-




Question Number 8050 by Chantria last updated on 28/Sep/16
The value of  ((18^3 +7^3 +3 ∙ 18 ∙ 7 ∙ 25)/(3^6 +6∙243∙2+15∙181∙4+20∙27∙8+15∙9∙16+6∙3∙32+64))  is
$$\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{18}^{\mathrm{3}} +\mathrm{7}^{\mathrm{3}} +\mathrm{3}\:\centerdot\:\mathrm{18}\:\centerdot\:\mathrm{7}\:\centerdot\:\mathrm{25}}{\mathrm{3}^{\mathrm{6}} +\mathrm{6}\centerdot\mathrm{243}\centerdot\mathrm{2}+\mathrm{15}\centerdot\mathrm{181}\centerdot\mathrm{4}+\mathrm{20}\centerdot\mathrm{27}\centerdot\mathrm{8}+\mathrm{15}\centerdot\mathrm{9}\centerdot\mathrm{16}+\mathrm{6}\centerdot\mathrm{3}\centerdot\mathrm{32}+\mathrm{64}} \\ $$$$\mathrm{is} \\ $$
Answered by sandy_suhendra last updated on 28/Sep/16
= (((18+7)^3 )/((3+2)^6 )) = ((25^3 )/5^6 ) = (5^6 /5^6 ) = 1  using :  (a+b)^3 =a^3 +b^3 +3ab(a+b)  (a+b)^6 =a^6 +6a^5 b+15a^4 b^2 +20a^3 b^3 +15a^2 b^4 +6ab^5 +b^6
$$=\:\frac{\left(\mathrm{18}+\mathrm{7}\right)^{\mathrm{3}} }{\left(\mathrm{3}+\mathrm{2}\right)^{\mathrm{6}} }\:=\:\frac{\mathrm{25}^{\mathrm{3}} }{\mathrm{5}^{\mathrm{6}} }\:=\:\frac{\mathrm{5}^{\mathrm{6}} }{\mathrm{5}^{\mathrm{6}} }\:=\:\mathrm{1} \\ $$$${using}\:: \\ $$$$\left({a}+{b}\right)^{\mathrm{3}} ={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +\mathrm{3}{ab}\left({a}+{b}\right) \\ $$$$\left({a}+{b}\right)^{\mathrm{6}} ={a}^{\mathrm{6}} +\mathrm{6}{a}^{\mathrm{5}} {b}+\mathrm{15}{a}^{\mathrm{4}} {b}^{\mathrm{2}} +\mathrm{20}{a}^{\mathrm{3}} {b}^{\mathrm{3}} +\mathrm{15}{a}^{\mathrm{2}} {b}^{\mathrm{4}} +\mathrm{6}{ab}^{\mathrm{5}} +{b}^{\mathrm{6}} \\ $$

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