Question: Find the value of x in : (a + x^1/2)^1/3 + (a - x^1/2)^1/3 = b^1/3
The expression is,
(a + x^1/2)^1/3 + (a - x^1/2)^1/3 = b^1/3 -- cube both sides
(a + x^1/2) + 3(a + x^1/2)^(2/3)*(a - x^1/2)^(1/3) + 3(a + x^1/2)^(1/3)*(a - x^1/2)^(2/3) + (a - x^1/2) = b
(2a) + 3(a + x^1/2)^(1/3)*(a - x^1/2)^(1/3){(a + x^1/2)^(1/3) + (a - x^1/2)^(1/3)} = b
3(a^2 - x)^(1/3){(a + x^1/2)^(1/3) + (a - x^1/2)^(1/3)} = b - 2a
{(a + x^1/2)^(1/3) + (a - x^1/2)^(1/3)} = (b - 2a)/{3(a^2 - x)^(1/3)}
Now substitute for b^1/3 = (a + x^1/2)^1/3 + (a - x^1/2)^1/3
b^1/3 = (b - 2a)/{3(a^2 - x)^(1/3)} -- cube both sides
b*(27(a^2 - x)) = (b - 2a)^3
27(a^2 - x) = (b - 2a)^3/b
x = a^2 - (1/27)(b - 2a)^(3)/b