**Problem: Prove that **

is a rational number.

**Proof:**

Let . We then have

We know that implies that , so we obtain

or,

Clearly, out of the roots of this equation is and the other two roots satisfy the equation which has no real solutions. (This equation can be found by polynomial division). Since is a real number, it follows that

, which is a rational number.

More later,

Nalin Pithwa

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*Related*

We can also cube the number(which is to be proved rational) , to get the same cubic.

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