# Solve x^sqrtx = 16 ?

Apr 6, 2017

$x = 4$

#### Explanation:

Making $y = \sqrt{x}$ we have

${\left({y}^{2}\right)}^{y} = {y}^{2 y} = {4}^{2}$ so ${y}^{y} = 4$

so $y = 2$

and $x = {y}^{2} = 4$

Now using the Lambert function $W$ we have

$y = \log \frac{4}{W \left(\log \left(4\right)\right)}$ and also

$x = {y}^{2} = {\left(\log \frac{4}{W \left(\log \left(4\right)\right)}\right)}^{2} = 4$