Why is this equation true?

Question

How is $e^{2x}$ equal to $e^x+e^{-x}$ ? I'm totally mad that I don't get it. Thanks.

Answer 1

that's not correct $$e^{2x}=e^xe^x=(e^x)^2$$ $$e^x+e^x=2e^x$$ $e^x+e^{-x}$ can't be simplified

$e^x+e^{-x}=e^{2x}$ can be true for a certain value of x but not $\forall \, x$

Answer 2

They are not equal.

$$e^{2\cdot 0}= 1\neq 2= e^0 + e^{-0}$$

We do have

$$e^{2x} = e^x \div e^{-x}$$

and I do have experience of mistook $\div$ as $+$.

Answer 3

If $e^{2x} = e^x+e^{-x}$, let $y = e^x$. Then $e^{2x} = y^2$ and $e^{-x} = 1/y$.

The equation becomes $y^2 = y+1/y$. Multiplying by $y$, this becomes $y^3 = y^2+1$.

This (according to Wolfy) has one real root (about $1.4656$) and two conjugate complex roots (about $-0.23279 \pm 0.79255 i$). The algebraic expressions are the usual cubic mess and do not seem interesting.

The $x$ corresponding to the real root is about $0.382245$.