The Easiest Path to Radical Math Success!

[@RyanLewis-Johnson-wq6xs]

X=Surd[1/(Sqrt[80]+Sqrt[76])+1/(Sqrt[76]+Sqrt[72])+1/(Sqrt[72]+Sqrt[68])+1/(Sqrt[68]+Sqrt[64]),3] X^7=(13Sqrt[5]-29)/2

[@ashokdubey8415]

The given equation transformed to x^3=sqrt(5)-2 and let
y^3=sqrt(5)+2 then
y^3-x^3=4 &
x^3.y^3=1 ,
Solving =>
sqrt(5)-1
x=----------------
2
x^7=1/2[13.sqrt(5)-29]

[@Quest3669]

X^3=√5-2 ; x^6= 9-4√5
x= (√5-1)/2>0 hence
x^7= (13√5-29)/2

[@Fjfurufjdfjd]

χ^7=[13(5)^(1/2)-29]/2

[@ManojkantSamal]

^=read as to the power
*=read as square root
Ans : X={(*5 -2)}^(7/3).....May be
Explain later.....

[@gregevgeni1864]

x = ³√(((√80-√76)+(√76-√72)+(√72-√68)+(√68-√64))/4) =
=³√((√80-√64)/4=³√((4√5-8)/4)=
=³√(√5-2) => x = ³√(√5-2) (1).
Too x³ = √5 -2 =>
x⁶ =(x³)² =(√5-2)²= 5-4√5+4= 9-4√5 (2) .
From (1) , (2) =>
x⁷ = x • x⁶ = (9-4√5)•³√(√5-2).
or x⁷ = (√5-2)⅓ • (√5-2)² = (√5-2)⅓+²
= (√5-2)⁷/³