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Question Number 132285 by bemath last updated on 13/Feb/21
Simplify the equation of (((x^(1/3) −x^(1/6) )(x^(1/2) +x)(x^(1/2) +x^(1/3) +x^(2/3) ))/((x^(4/3) −x)(x+x^(1/3) +x^(2/3) ))) with x ≠ 0
$$\mathrm{Simplify}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\: \\ $$$$\frac{\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}\right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\left(\mathrm{x}^{\frac{\mathrm{4}}{\mathrm{3}}} −\mathrm{x}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$\mathrm{with}\:\mathrm{x}\:\neq\:\mathrm{0} \\ $$$$ \\ $$
Answered by Olaf last updated on 13/Feb/21
X = (((x^(1/3) −x^(1/6) )(x^(1/2) +x)(x^(1/2) +x^(1/3) +x^(2/3) ))/((x^(4/3) −x)(x+x^(1/3) +x^(2/3) ))) Let a = x^(1/6) X = (((a^2 −a)(a^3 +a^6 )(a^3 +a^2 +a^4 ))/((a^8 −a)(a^6 +a^2 +a^4 ))) X = ((a(a−1)a^3 (a^3 +1)a^2 (a^2 +a+1))/(a(a^7 −1)a^2 (a^4 +a^2 +1))) X = ((a^3 (a−1)(a^3 +1)(a^2 +a+1))/((a^7 −1)(a^4 +a^2 +1))) X = ((a^3 (a−1)(a+1)(a^2 −a+1)(a^2 +a+1))/((a^7 −1)(a^4 +a^2 +1))) X = ((a^3 (a−1)(a+1)(a^4 +a^2 +1))/((a^7 −1)(a^4 +a^2 +1))) X = ((a^3 (a−1)(a+1))/(a^7 −1)) = ((a^3 (a^2 −1))/(a^7 −1)) X = ((x^(1/2) (x^(1/3) −1))/(x^(7/6) −1)) ...or X = ((a^3 (a+1))/(a^6 +a^5 +a^4 +a^3 +a^2 +a+1))
$$\mathrm{X}\:=\:\frac{\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}\right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\left(\mathrm{x}^{\frac{\mathrm{4}}{\mathrm{3}}} −\mathrm{x}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$\mathrm{Let}\:{a}\:=\:{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \\ $$$$\mathrm{X}\:=\:\frac{\left({a}^{\mathrm{2}} −{a}\right)\left({a}^{\mathrm{3}} +{a}^{\mathrm{6}} \right)\left({a}^{\mathrm{3}} +{a}^{\mathrm{2}} +{a}^{\mathrm{4}} \right)}{\left({a}^{\mathrm{8}} −{a}\right)\left({a}^{\mathrm{6}} +{a}^{\mathrm{2}} +{a}^{\mathrm{4}} \right)} \\ $$$$\mathrm{X}\:=\:\frac{{a}\left({a}−\mathrm{1}\right){a}^{\mathrm{3}} \left({a}^{\mathrm{3}} +\mathrm{1}\right){a}^{\mathrm{2}} \left({a}^{\mathrm{2}} +{a}+\mathrm{1}\right)}{{a}\left({a}^{\mathrm{7}} −\mathrm{1}\right){a}^{\mathrm{2}} \left({a}^{\mathrm{4}} +{a}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\mathrm{X}\:=\:\frac{{a}^{\mathrm{3}} \left({a}−\mathrm{1}\right)\left({a}^{\mathrm{3}} +\mathrm{1}\right)\left({a}^{\mathrm{2}} +{a}+\mathrm{1}\right)}{\left({a}^{\mathrm{7}} −\mathrm{1}\right)\left({a}^{\mathrm{4}} +{a}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\mathrm{X}\:=\:\frac{{a}^{\mathrm{3}} \left({a}−\mathrm{1}\right)\left({a}+\mathrm{1}\right)\left({a}^{\mathrm{2}} −{a}+\mathrm{1}\right)\left({a}^{\mathrm{2}} +{a}+\mathrm{1}\right)}{\left({a}^{\mathrm{7}} −\mathrm{1}\right)\left({a}^{\mathrm{4}} +{a}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\mathrm{X}\:=\:\frac{{a}^{\mathrm{3}} \left({a}−\mathrm{1}\right)\left({a}+\mathrm{1}\right)\left({a}^{\mathrm{4}} +{a}^{\mathrm{2}} +\mathrm{1}\right)}{\left({a}^{\mathrm{7}} −\mathrm{1}\right)\left({a}^{\mathrm{4}} +{a}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\mathrm{X}\:=\:\frac{{a}^{\mathrm{3}} \left({a}−\mathrm{1}\right)\left({a}+\mathrm{1}\right)}{{a}^{\mathrm{7}} −\mathrm{1}}\:=\:\frac{{a}^{\mathrm{3}} \left({a}^{\mathrm{2}} −\mathrm{1}\right)}{{a}^{\mathrm{7}} −\mathrm{1}} \\ $$$$\mathrm{X}\:=\:\frac{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{1}\right)}{{x}^{\frac{\mathrm{7}}{\mathrm{6}}} −\mathrm{1}} \\ $$$$…\mathrm{or}\:\mathrm{X}\:=\:\frac{{a}^{\mathrm{3}} \left({a}+\mathrm{1}\right)}{{a}^{\mathrm{6}} +{a}^{\mathrm{5}} +{a}^{\mathrm{4}} +{a}^{\mathrm{3}} +{a}^{\mathrm{2}} +{a}+\mathrm{1}} \\ $$
Answered by benjo_mathlover last updated on 13/Feb/21
((x^(1/6) (x^(1/6) −1)x^(1/2) (1+x^(1/2) )(x^(1/2) +x^(1/3) +x^(2/3) ))/(x(x^(1/3) −1)(x+x^(1/3) +x^(2/3) ))) = (((x^(1/6) −1)(1+x^(1/2) )(x^(1/2) +x^(1/3) +x^(2/3) ))/(x^(1/3) (x^(1/6) −1)(x^(1/6) +1)(x+x^(1/3) +x^(2/3) ))) = (((1+x^(1/2) )(x^(1/2) +x^(1/3) +x^(2/3) ))/(x^(1/3) (x^(1/6) +1)(x+x^(1/3) +x^(2/3) ))) = (((1+x^(1/2) )(x^(1/6) +1+x^(1/3) ))/((x^(1/6) +1)(x+x^(1/3) +x^(2/3) )))
$$\frac{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} −\mathrm{1}\right)\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{1}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\mathrm{x}\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{1}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$=\:\frac{\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} −\mathrm{1}\right)\left(\mathrm{1}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} −\mathrm{1}\right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} +\mathrm{1}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$=\:\frac{\left(\mathrm{1}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} +\mathrm{1}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$=\:\frac{\left(\mathrm{1}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} +\mathrm{1}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} \right)}{\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} +\mathrm{1}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$

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