+128460 I think they want you to evaluate the expression for x = 3√[7 + 5√2] - 1 / 3√[7 + 5√2]
This isn't as difficult as it looks.....just tedious.....first....let's just substitute a^(1/3) - a^(-1/3) in the given expression.....where a = 3√[7 + 5√2]
So we have
x^3 +3x - 14 for a^(1/3) - a^(-1/3) ....
[ a^(1/3) - a^(-1/3)]^3 +3 [a^(1/3) - a^(-1/3)] - 14 so we have
[a^(1/3)]^3 - 3[a^(1/3)]^2*[a^(-1/3)] + 3[a^(1/3)]*[a^(-1/3)]^2 - [a^(-1/3)]^3 + 3a^(1/3) - 3a^(-1/3) - 14 = [.........notice that the binomial expansion was used to expand the first term.......]
Simplify......
a - 3a^(1/3) + 3a^(-1/3) - a^(-1) + 3a^(1/3) - 3a^(-1/3) - 14 =
a - a^(-1) - 14 = (back-substitute for a = 3√[7 + 5√2] )
√[7 + 5√2] - ( √[7 + 5√2]) ^(-1) - 14 =
√[7 + 5√2] - 1 / √[7 + 5√2] - 14 = [rationalize the denominator in the second term ]
7 + 5√2 - 1[ 7 - 5√2] / [ (7 + 5√2) (7 - 5√2] - 14 =
7 + 5√2 - [ 7 - 5√2] / [ 49 - 50] - 14 =
7 + 5√2 - [7 - 5√2] / [-1] - 14 =
7 + 5√2 + [7 - 5√2] - 14 =
14 - 14 =
0 .........that was a lot of trouble just to get that....LOL!!!!
