Question Number 103047 by bemath last updated on 12/Jul/20

find the product of roots   ((2021))^(1/(3 ))  x^(log_(2021)  (x))  = x^3

Commented bymr W last updated on 12/Jul/20

Πx=2021^3  ?

Answered by floor(10²Eta[1]) last updated on 12/Jul/20

log_(2021) (x)=y⇒x=2021^y   ⇒((2021))^(1/3) .(2021^y )^y =(2021^y )^3   2021^(1/3) .2021^y^2  =2021^(3y)   ⇒y^2 +(1/3)=3y⇒y^2 −3y+(1/3)=0  product=x_1 .x_2 =2021^y_1  .2021^y_2    =2021^(y_1 +y_2 ) =2021^3

Answered by bemath last updated on 14/Jul/20

set x=2021^y  ⇔ y=log _(2021) (x)  ⇒2021^(1/3) (2021^y )^y  = 2021^(3y)   2021^(y^2 −3y+(1/3))  = 1   ⇒y^2 −3y+(1/3) = 0 ; Δ=9−(4/3)>0   ⇒(y−a)(y−b) = 0⇒a+b = 3  A=2021^a  ;B = 2021^b   ⇔A.B = 2021^(a+b)  = 2021^3   = 21^3  (mod 100) = 261 (mod  1000)