# Writing an algebraic expression as a single quotient

I have been working on this problem for a while and can't quite figure out how to do it. How can I turn this expression into a single quotient?

$\frac{2x}{(1-x)^\frac12} + \frac{2(1-x)^\frac{-1}{2}}{3}$

The first thing I did was move the negative exponent down in the denominator to obtain this expression.

$\frac{2x}{(1-x)^\frac12} + \frac{2}{3(1-x)^\frac{1}{2}}$

However, I'm stuck on how to combine it as a single quotient because the fractional exponents are throwing me off.

• Just factor out the $1/(1-x)^{1/2}$ from both terms. – Robert Israel Aug 24 '17 at 0:51

Hint: it may be more obvious if you work with $(1 - x)^{-1/2}$ in the numerators of the fractions.
Consider this as $2x(1-x)^{\frac{-1}{2}}+\frac{2}{3}(1-x)^{\frac{-1}{2}}$, then simply factor out $(1-x)^{\frac{-1}{2}}$ to obtain $(1-x)^{\frac{-1}{2}}(2x+\frac{2}{3})$. This is clearly $(1-x)^{\frac{-1}{2}}(\frac{6x+2}{3})$, then multiply it out to obtain one fraction, we have:
$\frac{6x+2}{3\sqrt{1-x}}=\frac{6x+2}{3(1-x)^{\frac{1}{2}}}$