# How do you simplify  (1+1/x)/(1/x)?

##### 1 Answer
Jun 16, 2016

When a fraction is in the denominator, you can treat it as multiplying by its reciprocal.

Recall that $\frac{1}{u} = {u}^{- 1}$. In that case, if we let $u = \frac{1}{x}$, then:

$\frac{1}{\left(\frac{1}{x}\right)}$

$= {\left(\frac{1}{x}\right)}^{- 1}$

$= \frac{1}{{x}^{- 1}}$

$= 1 \cdot {x}^{1}$

$= x$

So if you had been multiplying by $\frac{1}{\frac{1}{x}}$, you could instead multiply by $x$ to accomplish the same thing.

$\textcolor{b l u e}{\frac{1 + \frac{1}{x}}{\frac{1}{x}}}$

$= \left(1 + \frac{1}{x}\right) \cdot \frac{1}{\frac{1}{x}}$

$= \left(1 + \frac{1}{x}\right) \cdot {\left(\frac{1}{x}\right)}^{- 1}$

$= \left(1 + \frac{1}{x}\right) \cdot x$

$= 1 \cdot x + \frac{1}{\cancel{x}} \cdot \cancel{x}$

$= \textcolor{b l u e}{x + 1}$