jmc

Let the energy of a photon equal $E$ and the wavelength equal $\lambda$. Since the wavelength is inversely proportional to the energy, the product $E\lambda$ must be equal to some constant, say $k$. Given photons of red light with a wavelength of $7\times 10^{-7}$, we can write: $$\begin{array}{rl}E(7\times 10^{-7})& =k\\ \Rightarrow 7\times 10^{-7}& =\frac{k}{E}\end{array}$$ Now, we are asked to find the wavelength of a photon with 2000 times the energy of red light. Substitute $2000E$ for $E$ in the original expression: $$\begin{array}{rl}(2000E)\lambda & =k\\ \Rightarrow \lambda & =\frac{k}{2000E}\\ & =\frac{1}{2000}\cdot \frac{k}{E}\\ & =\frac{1}{2\times 10^3}\cdot 7\times 10^{-7}\\ & =3.5\times 10^{-10}\text{ meters}\end{array}$$ Therefore, we have $a+b=\boxed{-6.5}$.