Why is \( e^{C} = Ce \) but \( e^{x^2+C} = De^{x^2} \)?

[vanceEE]

Why does $$ e^{C} = Ce ?$$

 

[1MileCrash]

Why does $$ e^{C} = Ce ?$$

It doesn't..

If C is an arbitrary constant, it is true that I can say:

$$ e^{C} = e * e^{C-1} = C_{2}e $$

Because $$ e^{C-1} $$ is just some other constant. But it is not the same as C, it is a new constant.

 

[vanceEE]

It doesn't..

But $$e^{x^2+C} = De^{x^2} $$ are we just considering e^C to be an arbitrary constant since it is a number multiplied by a number?

 

[vanceEE]

It doesn't..

If C is an arbitrary constant, it is true that I can say:

$$ e^{C} = e * e^{C-1} = C_{2}e $$

Because $$ e^{C-1} $$ is just some other constant. But it is not the same as C, it is a new constant.

Ok thanks

 

[Mark44]

But $$e^{x^2+C} = De^{x^2} $$ are we just considering e^C to be an arbitrary constant since it is a number multiplied by a number?

Probably already answered, but here are the details.
e^{x2 + C} = e^x2 * e^C = D e^x2, where D = e^C.

Your question is really about the properties of exponents, and not specifically about the natural number e. The basic property here is a^{m + n} = a^m * aⁿ.