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? 

If you expand $(1+x)^k$ you get $1^k+k1^{k-1}x+{k \choose 2}1^{k-2}x^2+\dots =1+kx +\frac 12k(k-1)x^2+\dots $  If $kx \ll1$ the following terms are small.
