approximately equal derivation

[REcr]

Im in a grad level fluid mechanics class and I came across this approximation (1+kasin(kx))^-6 ≈ (1-6kasin(kx)) for this problem k<<1 so i understand the higher order terms can be dropped out for simplification but where is the -6 coming from? I have never in my life seen anything like this and i'm completely dumbfounded, please help shed some light.

 

[Ishuda]

Im in a grad level fluid mechanics class and I came across this approximation (1+kasin(kx))^-6 ≈ (1-6kasin(kx)) for this problem k<<1 so i understand the higher order terms can be dropped out for simplification but where is the -6 coming from? I have never in my life seen anything like this and i'm completely dumbfounded, please help shed some light.

That is just the binomial expansion dropping higher order terms
(1 + t)ⁿ = 1 + n t + \(\displaystyle \frac{1}{2}\) n (n-1) t² + ...
t = ka sin(kx)
n = - 6

 

[REcr]

Awesome! thank you!