A Tricky Radical Problem | Math Olympiad Preparation

[@RyanLewis-Johnson-wq6xs]

Sqrt[Cbrt[28]-Cbrt[27]]=0.3 recurring (Cbrt[98]-Cbrt[28]-1)

[@RyanLewis-Johnson-wq6xs]

Sqrt[Cbrt[28]-Cbrt[27]]=(Cbrt[98]-Cbrt[28]-1)/3

[@RyanLewis-Johnson-wq6xs]

Sqrt[Cbrt[28]-Cbrt[27]]=Sqrt[-3+Cbrt[28]]

[@walterwen2975]

Math Olympiad Preparation: √(³√28 - ³√27) = ​​?
Let: x = ³√28 - ³√27 = ³√28 - 3 > 0; x + 3 = ³√28, (x + 3)³ = (³√28)³
x³ + 9x² + 27x + 27 = 28, x³ + 9x² + 27x = 1
Trial-and-error solution:
x³ + 9x² + 27x = 1; 1/27 > x > 0, 1 > 27x > 9x² > x³
x =1/27 = 0.037: (1/27)³ + 9(1/27)² + 27(1/27) = 1.0114; 1/27 > x
x = 1/28 = 0.0357: (1/28)³ + 9(1/28)² + 27(1/28) = 0.9758 < 1; 1/27 > x > 1/28
x = 0.036: 0.036³ + 9(0.036²) + 27(0.036) = 0.9837 < 1; 0.037 > x > 0.036
x = 0.0366: 0.0366³ + 9(0.0366²) + 27(0.0366) = 1.0003; Proved
x = ³√28 - ³√27 = 0.0366; √(³√28 - ³√27) = √x = √0.0366 = 0.191
The calculation was achieved on a smartphone with a standard calculator app
Answer check:
x = √(³√28 - ³√27) = 0.191; Confirmed as shown
Final answer:
√(³√28 - ³√27) = 0.191

[@RyanLewis-Johnson-wq6xs]

Sqrt[Cbrt[28]-Cbrt[27]]=Sqrt[Cbrt[28]-3] final answer

[@RealQinnMalloryu4]

(x^3)^2 ➖ (28)^2 ➖ (x^3)^3(27)^2= {x^9 ➖ 584} ={x^9 ➖ 729}= { 575 ➖ 720}= 255 10^20^55 10^20^5^11 2^5^5^4^5^11^1 1^1^12^2^1^1^1 1^2^1^1 21;(x ➖ 2x+1).