How to simplify √x when used as exponent?

Question

Could anyone help me understand how to simplify the following expression.

$$x^\sqrt x$$

If there was a number instead of $\sqrt x$ as an exponent, it wouldn't be a problem for me. But I have never seen $\sqrt x$ as an exponent before.

Here's the original question:

If $$x^{x^\sqrt x} = (x\sqrt x)^x,$$ what's the value of $x$?

Please NOTICE that $\sqrt x$ on the LEFT is an exponent of exponent.

Answer 1

Hint:

$$x\sqrt{x}=x^{\frac32}$$

Thus, you have: $$x^{(x^\sqrt x)}=x^{\frac32x}$$

which gives $$x^\sqrt x=\frac32x\tag1$$

Also don't forget the trivial solution $x=1$

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Further hint:

You can rearrange $(1)$ to $$x=(\log_x(\frac32)+1)^2$$ and, since we have isolated an $x$ term, we can iterate:

$$x_{k+1}=(\log_{x_k}(\frac32)+1)^2$$

Set $x_0=\frac32$ and you will see this converges to $\frac 94$

Can you isolate another $x$ term in $1$ to gain another iteration formula?

Answer 2

To simplify $x^{\sqrt{x}}$, you can try $\exp(\sqrt{x} \ln x)$.

Answer 3

Hint:

Write $x\sqrt x= x^{3/2} $

Hence, $x^{x^\sqrt x}=x^{3x/2} \Rightarrow x^\sqrt x=3x/2 \Rightarrow x^{\sqrt x-1} =\frac{3} {2} \Rightarrow x=9/4$ is one of the solutions.