## Notation Obfuscation

As I was walking to my office, a student flagged me down and asked how to solve a problem similar to this,  $4(1-x)^\frac{-1}{2}+\sqrt{1-x}=0$. He  could not understand the steps as written in his student’s solution manual.

I blurted out, “That’s disgusting!”  He agreed though probably not for the same reason and then I showed him what was going on.  The problem as written is  notationally disgusting.  It mixes fractional exponent notation and root notation.  This is dead wrong.  Either write  $4\frac{1}{\sqrt{1-x}}+\sqrt{1-x}=0$ or $4(1-x)^\frac{-1}{2}+(1-x)^\frac{1}{2}=0$.  Solve the first equation by multiplying each term by $\sqrt{1-x}$ and the second equation by multiplying each term by $(1-x)^\frac{1}{2}$.  The solution manual multiplied the original equation by $(1-x)^\frac{1}{2}$ expecting the student to know that $(1-x)^\frac{1}{2}\sqrt{1-x}$ is $(1-x)$.  In the student’s problem the answer satisfied the implicit domain restriction, here $x < 1$.   Notice that for our problem the  “solution” $x = 5$ is outside of this domain and the problem has no real number solution. (Give this problem to Wolfram|Alpha and it gives $x = 5$.

We need to make up our mind.  Use  root notation or  rational exponent notation, certainly not both in the same expression.  I am in favor of using rational exponent exclusively and teaching that for $x^\frac{1}{2}$   $x$ must be non-negative when  dealing with real numbers.  If we do this,  many of the special rules for working with radicals will go away.  Root notation is archaic.  Let’s teach it  as an historical artifact only and save our students time, trouble, and tedium.