Question

How do you simplify `(2/a)/(1/(a+6))`?

Answer 1

`(2/a)/(1/(a+6)) = 2+12/a`

with exclusion `a != -6`

Explanation:

Note that division is the same as multiplication by the reciprocal:

`(2/a)/(1/(a+6)) = (2/a)-:(1/(a+6))`

`color(white)((2/a)/(1/(a+6))) = (2/a)*((a+6)/1)`

`color(white)((2/a)/(1/(a+6))) = (2a+12)/a`

`color(white)((2/a)/(1/(a+6))) = 2+12/a`

Note that both the original expression and the simplified expression are undefined when `a=0`, since they both involve division by `0`.

By way of contrast, the original expression is undefined when `a=-6`, but the simplified expression is well defined.

So the original expression and simplified expression are not equivalent when `a=-6`. So we need to explicitly exclude the value `a=-6`.