# single algebraic fraction

• Aug 11th 2008, 12:27 PM
comet2000
single algebraic fraction
Write each expression as a single algebraic fraction.
(1-xy^-1)^-1
• Aug 11th 2008, 12:33 PM
Chop Suey
$\displaystyle (1-(xy)^{-1})^{-1}$

Remember:
$\displaystyle a^{-1} = \frac{1}{a}$

So:
$\displaystyle (1-(xy)^{-1})^{-1} = \frac{1}{1-(xy)^{-1}} = \frac{1}{1-\frac{1}{xy}}$

Combine 1 and $\displaystyle \frac{1}{xy}$. Remember that $\displaystyle 1 = \frac{xy}{xy}$.

$\displaystyle \frac{1}{1-\frac{1}{xy}} = \frac{1}{\frac{xy}{xy}-\frac{1}{xy}} = \frac{1}{\frac{xy - 1}{xy}}$

Now, this expression can be rearranged like this:

$\displaystyle \frac{1}{\frac{xy - 1}{xy}} = 1 \cdot \frac{xy}{xy-1} = \frac{xy}{xy-1}$
• Aug 11th 2008, 01:16 PM
wingless
$\displaystyle xy^{-1}$ can mean $\displaystyle \frac{x}{y}$ too.
• Aug 11th 2008, 01:33 PM
Chop Suey
If you meant $\displaystyle xy^{-1}$ as Wingless pointed out:

$\displaystyle (1-xy^{-1})^{-1}$

$\displaystyle = \frac{1}{1-\frac{x}{y}}$

$\displaystyle = \frac{1}{\frac{y-x}{y}}$

$\displaystyle = 1 \cdot \frac{y}{y-x} = \frac{y}{y-x}$