# Proof that #N = (45+29 sqrt(2))^(1/3)+(45-29 sqrt(2))^(1/3)# is a integer ?

##### 2 Answers

Consider

This has one Real root which is

#### Explanation:

Consider the equation:

#t^3-21t-90 = 0#

Using Cardano's method to solve it, let

Then:

#u^3+v^3+3(uv-7)(u+v)-90 = 0#

To eliminate the term in

Then:

#u^3+7^3/u^3-90 = 0#

Multiply through by

#(u^3)^2-90(u^3)+343 = 0#

by the quadratic formula, this has roots:

#u^3 = (90+-sqrt(90^2-(4*343)))/2#

#color(white)(u^3) = 45 +- 1/2sqrt(8100-1372)#

#color(white)(u^3) = 45 +- 1/2sqrt(6728)#

#color(white)(u^3) = 45 +- 29sqrt(2)#

Since this is Real and the derivation was symmetric in

#t_1 = root(3)(45+29sqrt(2))+root(3)(45-29sqrt(2))#

but we find:

#(6)^3-21(6)-90 = 216 - 126 - 90 = 0#

So the Real zero of

So

**Footnote**

To find the cubic equation, I used Cardano's method backwards.

#### Explanation:

Making

so

or calling

with