Solve for x: 2^x + 6^x + 16^x = 7^x + 8^x + 9^x.
Is there an algebraic way to solve for x? My way of solving this is through graphing.
Well I'd have to say that just by looking at it, 0 is a solution:
$\displaystyle 2^0 + 6^0 + 16^0 = 7^0 + 8^0 + 9^0$
$\displaystyle 1 + 1 + 1 = 1 + 1 + 1$
$\displaystyle 3 = 3$
Also 1 is a solution since all those numbers add to 24:
$\displaystyle 2^1 + 6^1 + 16^1 = 7^1 + 8^1 + 9^1$
$\displaystyle 24 = 24$
However, beyond that, I do not know. There might be more than 2 solutions, but maybe whoever gave you this problem didn't think of that. I'd say look at the solutions of 1 and 0 if you have a problem like this, and go from there. In my opinion this looks like a thinking problem, something you're just suppose to see.
I think you already found all the integer solutions, based on my graph. I did not see well as you did. I thought it needs higher analysis . . . . i lacked insight on that, SEEING what is the obvious.
Actually, i found that problem in one website a year ago, that i can no longer locate. It intrigues me, so i copied it in one of my notes.