Let x be a non-zero real number such that (x³+20x)^1/5=(x⁵-20x)^1/3. Find the product of all possible values of x.

First the short way.  The apparent solutions from graphing the equations are x=0 and x=±√5
So the product of the non-zero solutions is -5

I wanted to confirm this so I raised both sides to the 15th power, then simplified the result to a 22nd degree polynomial with all even terms.

Using the substitution y=x² brings us to a 10th degree polynomial (too big to type out here)
I was able to use synthetic division to show y=5 is a zero of the polynomial so x²=5 and so indeed x=±√5

I was also able to find the some non-real solutions y=x²=-4 so x= ±2i and also an irrational y=x²≈.0024990669

Posted by Jer on 2014-02-23 21:35:56