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Find-61-3-24-3-61-3-37-3-




Question Number 208469 by hardmath last updated on 16/Jun/24
Find:     ((61^3   +  24^3 )/(61^3   +  37^3 ))  =  ?
$$\mathrm{Find}:\:\:\:\:\:\frac{\mathrm{61}^{\mathrm{3}} \:\:+\:\:\mathrm{24}^{\mathrm{3}} }{\mathrm{61}^{\mathrm{3}} \:\:+\:\:\mathrm{37}^{\mathrm{3}} }\:\:=\:\:? \\ $$
Answered by Frix last updated on 16/Jun/24
a=61; b=24; a−b=37  a^3 +b^3 =(a+b)(a^2 −ab+b^2 )  a^3 +(a−b)^3 =  =(a+(a−b))(a^2 −a(a−b)+(a−b)^2 )=  =(2a−b)(a^2 −ab+b^2 )  ⇒ ((a^3 +b^3 )/(a^3 +(a−b)^3 ))=((a+b)/(2a−b))  ((61+24)/(2×61−24))=((85)/(98))
$${a}=\mathrm{61};\:{b}=\mathrm{24};\:{a}−{b}=\mathrm{37} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\left({a}+{b}\right)\left({a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{3}} +\left({a}−{b}\right)^{\mathrm{3}} = \\ $$$$=\left({a}+\left({a}−{b}\right)\right)\left({a}^{\mathrm{2}} −{a}\left({a}−{b}\right)+\left({a}−{b}\right)^{\mathrm{2}} \right)= \\ $$$$=\left(\mathrm{2}{a}−{b}\right)\left({a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\:\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} }{{a}^{\mathrm{3}} +\left({a}−{b}\right)^{\mathrm{3}} }=\frac{{a}+{b}}{\mathrm{2}{a}−{b}} \\ $$$$\frac{\mathrm{61}+\mathrm{24}}{\mathrm{2}×\mathrm{61}−\mathrm{24}}=\frac{\mathrm{85}}{\mathrm{98}} \\ $$
Commented by hardmath last updated on 17/Jun/24
thankyou dear professor
$$\mathrm{thankyou}\:\mathrm{dear}\:\mathrm{professor} \\ $$

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