What is the PEMDAS for $e^{x^2}$

Question

$e^{x^2}$

Is this read as $(e^x)^2$ or $e^{(x^2)}$

Why does one exponent take precedence over the other? Is it because of sequential ordering of the same operation?

Answer 1

Based on the laws of exponents, $$(e^x)^2=e^{(2x)}$$ Hence, there is no benefit of interpreting it this way, and instead we interpret $$e^{x^2}=e^{(x^2)}$$

Answer 2

It’s not PEMDAS so much as it is a choice of parenthesization. This is more a matter of convention than anything else. But as @vadim123 says, the wrong parenthesization would be relatively useless.

Answer 3

It reads $$e^{(x^2)}$$ the other is $$({e^x})^2=e^{2x}$$

Answer 4

For this expression, it is implicit that we mean $e^{(x^2)}$ due to the nature of function composition, that is, the entirety of $x^2$ is present as the power of $e,$ without any parentheses.

What you can see is that if $f(x)=e^x,$ and $g(x)=x^2,$ then $f(g(x))=e^{x^2}.$

However, it is important to note that $f(g(x))\ne g(f(x))$ in general. So we can see that $g(f(x))=(e^x)^2=e^{2x}.$

So PEMDAS would say evaluate parentheses, and this means to evaluate $x^2,$ then exponents, so whatever the value of $x^2$ is evaluate this as the power of $e.$