Why is $\sqrt x\ne\frac1{x^2}$? Which step is illegal? 1. $\sqrt x=x^{\frac12}$ 2. $\frac12=2^{-1}$ 3. $x^{\frac12}= x^{2^{-1}}$ 4. $a^{b^c}=a^{bc}$ 5. $x^{2^{-1}}=x^{-2}$ 6. $a^{-b}=\frac1{a^b}$ 7. $x^{-2}=\frac1{x^2}$ Am I correct in assuming that it is step 3, because $a^{b^c}$ is not the same as $a^d$ where $d=b^c$ but rather $a^d=a^{(b^c)}$?

For sake of natation:

$(a^b)^c=a^{bc} $

But $a^{(b^c)}\
e a^{bc} $

So $x^{\frac 12}=x^{(2^{-1})}\
e (x^2)^{-1}=\frac 1 {x^2} $.

Others explained but I hope notation makes it clear.