Peter solved the equation $$2^{-x}=8$$ this way:
(1) He modified the left side of the equation: $$-2^x=8$$
(2) Then he converted number $8$ to the power with the base $2$: $$-2^x=2^3$$
(3) From the equality of the bases, he deduced: $$ \begin{gather} -x=3 \cr x=-3 \end{gather} $$
Then he did a check: $$L=2^{-\left(-3\right)}=2^3=8;~R=8;~L=R$$ The teacher gave Peter for his solution an insufficient grade. Petr asked his classmates for comments. Which one is correct?
Sandra claims that Peter made an error in steps (1) and (3).
John is convinced that the teacher is wrong because he did not notice that the check turned out well.
Bill thinks that Peter made an error in step (1). All other steps are correct.
Richard is convinced that the mistake is in the assignment. In the exponential equations cannot be a negative value in the exponent.
The equation can be correctly solved as: $$ \begin{gather} 2^{-x}=8 \cr 2^{-x}=2^3 \cr -x=3 \cr x=-3 \end{gather} $$ There are two errors in Peter’s solution. The first one is in step (1) because the equality $2^{-x}=-2^x$ does not hold. The second one is in step (3). From the equality $-2^x=2^3$ we cannot deduce that $-x=3$. The fact that the check turned out well does not say anything about correctness of our solution.