How to transform an exponential function

Question

I am stuck on a line in a problem. How do I get from

$4-2^{j+1}+4-2^{j-1}$

to

$8-5\times2^{j-1}$

please.

Your help would be greatly appreciated. What technique is it please so I can learn it?

Thanks

Answer 1

$4-2^{j+1}+4-2^{j-1}=8-2^{j+1}-2^{j-1}=8-2^{j-1+2}-2^{j-1}=8-4\cdot2^{j-1}-2^{j-1}=8-5\cdot2^{j-1}$

Answer 2

By the law of exponent addition,

$$2^{j+1}+2^{j-1}=2^j2^1+2^j2^{-1}=2^j(2^1+2^{-1}).$$

You can also write

$$2^{j+1}+2^{j-1}=2^{j-1}2^2+2^{j-1}2^0=2^{j-1}(2^2+1).$$