Using $(1+x)^k \approx 1+kx$ to approximate? 
Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. 

I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does $(1+x)^k \approx 1+kx$?  
$x^k$ is $x \times x \times x \times \dots x\;\;$ a certain $k$ number of times. Where do they get $kx$ from? 

One can expand the function $(1 + x)^{\alpha}$ in a Taylor series called the binomial series. It takes on the form
$$(1 + x)^{\alpha} = 1 + \alpha x + \frac{\alpha (\alpha - 1)}{2!} x^2 + \frac{\alpha (\alpha - 1)(\alpha - 2)}{3!} x^3 + \dots$$
Now if $x$ is small and $\alpha$ isn't too large relative to $x$, all the high-order terms can be neglected to give the first order approximation you're asking about.
