Jameson Lucero

2022-07-07

algebra exponents and fractions
I could be over thinking or tired... But I am to embarrassed to ask my prof. this probably very simple algebra rule I am ignorant of... Also this is just a snip-it from a inductive proof example.
Say you have something like this
$\frac{{4}^{k}-1}{3}+\frac{3\ast {4}^{k}}{3}=$
$\frac{4\ast {4}^{k}-1}{3}$
My question is what algebra rules or rules of fractions allow this? In other words what computation is going on? I can agree that the 3 should cancel out
$\frac{3\ast {4}^{k}}{3}={4}^{k}$
But wouldn't that mean it be something like this
$\frac{\left({4}^{k}-1\right)\ast {4}^{k}}{3}$
or more accurately
$\frac{\left({4}^{k}-1\right)}{3}\ast {4}^{k}$
and not this?
$\frac{4\ast {4}^{k}-1}{3}$

jugf5

Expert