Mathematics - Factoring Fractional Exponents

Me and another classmate are debating the true answer to the following equation. This is a college algebra question.

x^(3/2) + 8x^(1/2) + 16x^(-1/2) = 0

The book says no solution.

My classmate says it has a solution.

I feel the negative exponent will prevent it from being solved. Is this correct, or is there more to it?

If it can be solved, can you show me the proof.

**<edit 1> sorry edited out his proof on accident. i will put it back in tomorrow. </edit 1>**