Question Number 148257 by liberty last updated on 26/Jul/21

Answered by Olaf_Thorendsen last updated on 26/Jul/21

(2^x −4)^3 +(4^x −2)^3 = (4^x +2^x −6)^3 Let u = 2^x (u−4)^3 +(u^2 −2)^3 = (u^2 +u−6)^3 (u−4)^3 +(u^2 −2)^3 = [(u^2 −2)+(u−4)]^3 0 = 3(u^2 −2)^2 (u−4)+3(u^2 −2)(u−4)^2 (u^2 −2)(u−4)(u^2 −2+u−4) = 0 (u+(√2))(u−(√2))(u−4)(u+3)(u−2) = 0 u = ±(√2), 4,−3, 2 −(√2), −3 impossible because u = 2^x >0 u = (√2), 2, 4 ⇒ x = (1/2), 1, 2 S = {(1/2), 1, 2}

Answered by Rasheed.Sindhi last updated on 26/Jul/21

(2^x −4_(u) )^3 +(4^x −2_(v) )^3 =(4^x +2^x −6)^3 u^3 +v^3 =(u+v)^3 u^3 +v^3 =u^3 +v^3 +2uv(u+v) uv(u+v)=0 u=0 ∣ v=0 ∣ u+v=0 2^x −4=0 ∣ 4^x −2=0 ∣ 2^x −4=−4^x +2 2^x =2^2 ∣ 2^(2x) =2^1 ∣ 2^x _(y) +2^(2x) −6=0 x=2 ∣ 2x=1 ∣ y^2 +y−6=0 ∣ x=1/2 ∣ y^2 +y−6=0 ∣ (y+3)(y−2)=0 ∣ y=−3 ∣ y=2 ∣ 2^x =−3 ∣ 2^x =2^1 ∣not real ∣ x=1 2^x =−3 xlog(2)=log(−3) x=((log(−3))/(log(2))) =1.585+4.532 i x=1, 2 , 1/2 , 1.585+4.532 i