A Nice Radical Equation | An Algebra Challenge

[@RashmiRay-c1y]

Let y=x^1/3+2 > 65 -(y-2)^6 = (1-y)^6 > 2y^6-18y^5+75y^4-180y^3+255y^2-198y=0. Thus, y=0 is a solution, which means x = -8.

[@RajeshKumar-wu7ox]

x=-8, 2x^5+2t^4+11t^3-2t^2+19t-32=0

[@RyanLewis-Johnson-wq6xs]

Surd[65-(-8)^2,6]+Surd[-8,3]+1=0

[@Quest3669]

X= -8

[@davidshen5916]

Let A=X^（1/3）， B=（65-X^2）^（1/6）， B+A+1=0， B^6+A^6=（65-X^2）+X^2=65

[@kassuskassus6263]

x=-8.

[@RealQinnMalloryu4]

(x^6 )^2➖( 65x^6)^2(x^2)^2+x^3+1=0 =x^12=4225 {x^12➖ 4225}= 4213=x^4 {4213 ➖ x^4}= 4109 {x^3+x3 ➖}=x^6 {1+1 ➖}=2 {x^6+2}= =2x^6 {4109+2x^6}=4111x^6 2111^1111 2^1^1^1x^6 2x^3^2 1^1x^3^2 x^3^2 (x ➖ 3x+2).

[@RyanLewis-Johnson-wq6xs]

X=-8

[@abcekkdo3749]

X=-8