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

Feb 27, 2017

$\frac{\frac{2}{a}}{\frac{1}{a + 6}} = 2 + \frac{12}{a}$

with exclusion $a \ne - 6$

Explanation:

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

$\frac{\frac{2}{a}}{\frac{1}{a + 6}} = \left(\frac{2}{a}\right) \div \left(\frac{1}{a + 6}\right)$

$\textcolor{w h i t e}{\frac{\frac{2}{a}}{\frac{1}{a + 6}}} = \left(\frac{2}{a}\right) \cdot \left(\frac{a + 6}{1}\right)$

$\textcolor{w h i t e}{\frac{\frac{2}{a}}{\frac{1}{a + 6}}} = \frac{2 a + 12}{a}$

$\textcolor{w h i t e}{\frac{\frac{2}{a}}{\frac{1}{a + 6}}} = 2 + \frac{12}{a}$

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 $a = - 6$.