Locomotivation

$\displaystyle v = k\sqrt{x}$

$\displaystyle \Rightarrow v^2 = k^2x$

And Eq. of motion $\displaystyle v^2 = v_0^2 + 2ax$

$\displaystyle \Rightarrow a = \frac{k^2}{2},~~~v_0 = 0$

$\displaystyle s = \frac{1}{2}at^2 = \frac{k^2 t^2}{4}$

$\displaystyle \Rightarrow W_{done} = \frac{mk^2\cdot k^2t^2}{8} = \frac{mk^4t^2}{8}$

$\displaystyle \boxed{\therefore~ 7}$

I hope this is the easiest method! (Well I used Calculus!)

I'm not going to give a very complicated solution which will involve math. I'm going to relay the concepts required to solve the problem, so others can read up on those concepts and make full use of them in solving the problem.

The first thing that must be done is to define the relationship between the velocity and displacement. Now, it is clear that this problem requires differentiation and integration and so, the first thing that should be done is to differentiate the velocity to obtain an expression for the acceleration. Before this, it should be quite obvious that v is equal to some constant multiplied by the square root of the velocity, which is equal to the first derivative of x with respect to time. Now, we have obtained a differential equation which will make it easy for us to obtain quantities which will help us later on in the problem

What you will find is that the acceleration is a constant, This is really neat, isn't it? Now, the work done is simply the integral F. dx. But wait, a minute, we do not have an expression for dx, right?

Actually, we do. dx is simple equal to v dt, where v has already been given as a function of x. The aim now, is to find v as a function of time, so that we can do the integration F.v dt. Finding v as a function of time is simply a matter of solving a simple differential equation or finding a solution online. Because the force is a constant by itself, what we end up with, eventually, is the constant c multiplied by the integral of t.dt., which is equal to the work done by the force. Solving the integral, we find that m has an exponent of 1, the constant k, which relates the velocity and position, has an exponent of 4 and the time has an exponent of 2. Adding them gives us the answer 7.

I try, very much, not to include a lot of mathematical notation in these types of solutions, simply because I do not want to confuse those who are simply starting out on doing physics. I also want to emphasize that this was my thought process as I was solving the problem. What matters here are the concepts. The equations simply relay the concepts in an easy to understand way but it is a skill to relay the concepts without the equations. Anybody who has solved the problem is welcome to put in numerical solution to the problem :)