Question about dividing simple fractions

Question

I had some exercises and I came upon something that got me confused.

$$\frac{6*10^{w}}{7}$$ Turns out to be $$\frac{6}{7}*\frac{10^{w}}{7}$$ $$\frac{6*10^{w}}{49}$$ On the other hand, for example:$$\frac{12*10^{w}}{14}$$ does NOT equal: $$\frac{2*6}{2*7}*\frac{10^{w}}{7}$$ $$\frac{6}{7}*\frac{10^{w}}{7}$$ $$\frac{6*10^{w}}{49}$$ Why is that? I'm a bit confused so if I could get cleared up on some rules I've missed, I would appreciate it. Thanks.

Answer 1

$\frac{6*10^{w}}{7}$ is not equal to $\frac{6}{7}*\frac{10^{w}}{7}$, it is equal to $\frac{6}{1}*\frac{10^{w}}{7}$ or $\frac{6}{7}*\frac{10^{w}}{1}$ though.

It is not like a summation.

Generally, we have:

$\frac{a+b}{c}=\frac ac + \frac bc$

and

$\frac{ab}{c}=\frac a1 * \frac bc=\frac ac * \frac b1$

Answer 2

Note that $$\frac{6*10^{w}}{7} \ne \frac{6}{7}*\frac{10^{w}}{7}$$

Because you have only one $7$ on the left side but two $7$ on the right side.

cross out the extra $7$ and you will not be confused any more.