Answer
$k = 2\sqrt{e}$
Work Step by Step
If $e^{2x} = k~\sqrt{x}$ has one solution, then the curves $y = e^{2x}$ and $y = k~\sqrt{x}$ have one point of intersection and the two curves are tangent to each other at this point.
If the two curves intersect at one point:
$e^{2x} = k~\sqrt{x}$
$k = \frac{e^{2x}}{\sqrt{x}}$
If the two curves are tangent to each other at one point, then the derivatives are equal at this point:
$2e^{2x} = \frac{k}{2\sqrt{x}}$
$k = 4~\sqrt{x}~e^{2x}$
We can equate the two expressions for $k$ to find the value of $x$ where the curves are tangent to each other:
$k = 4~\sqrt{x}~e^{2x} = \frac{e^{2x}}{\sqrt{x}}$
$x = \frac{1}{4}$
We can find the value of $k$:
$e^{2x} = k~\sqrt{x}$
$e^{2(\frac{1}{4})} = k~\sqrt{\frac{1}{4}}$
$k = 2e^{1/2}$
$k = 2\sqrt{e}$