Exchanging powers

• I had a misunderstanding about the power of a power ${a^b}^c$.

• $(a^b{)}^c=a^{bc}=(a^c{)}^b$ is correct.

  • I misunderstood this and mistakenly believed that $a^{(b^c)}=a^{bc}=a^{(c^b)}$ (which is incorrect).

• By the way,

  • When the exponents are stacked, the calculation starts from the topmost power. Operator Precedence - Wikipedia

  • I didn’t know this either.
    • (I actually thought it didn’t matter which way you calculated it)
  • When we have $e^{x^2}$, it means $e^{(x^2)}$.
    • So, $(e^x{)}^2=e^{2x}$ is true, but $e^{x^2}\ne e^{2x}$.