Question:
Photons of red light have a wavelength of approximately $7\times 10^{-7}$ meters. The energy of a photon is inversely proportional to its wavelength. A photon with 2000 times the energy as a photon of red light will have a wavelength that can be written as $a\cdot 10^b$ meters, where $1\le a < 10$. (In other words, in scientific notation.)  What is $a+b$ written as a decimal?

Answer:
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 $7\times10^{-7}$, we can write: \begin{align*}
E(7\times10^{-7})&=k\\
\Rightarrow\qquad 7\times10^{-7}&=\frac{k}{E}
\end{align*} 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{align*}
(2000E)\lambda&=k\\
\Rightarrow\qquad \lambda&=\frac{k}{2000E}\\
&=\frac{1}{2000}\cdot\frac{k}{E}\\
&=\frac{1}{2\times10^3}\cdot7\times10^{-7}\\
&={3.5\times10^{-10} \text{ meters}}
\end{align*} Therefore, we have $a+b = \boxed{-6.5}$.