Subtracting 2 fractions with variables in the denominator that have different exponents.

Question

Sorry for the relatively elementary question, but I am having trouble remembering exactly how to do this type of problem. I am looking to simplify this: $$ \frac{3}{4t^{1/4}} - \frac{1}{2t^{3/4}} $$

and it needs to simplify to this:

$$ \frac{3\sqrt{t} - 2}{4 t^{3/4}} $$

Thanks!

K

Answer 1

In general to find the common denominator of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ take the $lcm(b,d)$. If that step is computationally cumbersome, then you can always take $bd$ as a common denominator.

That is to say, $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$.

Answer 2

You can do something like: \begin{eqnarray} \frac{3}{4t^{1/4}}-\frac{1}{2t^{3/4}} &=& \frac{3}{4t^{1/4}}\frac{t^{2/4}}{t^{2/4}} -\frac{1}{2t^{3/4}}\frac{2}{2} \\ &=&\frac{3 t^{2/4}}{4t^{3/4}} - \frac{2}{4t^{3/4}} \\ &=&\frac{3 t^{1/2}-2}{4t^{3/4}} \\ &=&\frac{3 \sqrt{t}-2}{4t^{3/4}} \end{eqnarray}

The point is that you make the denominators same and then you can simply add the numerators that turn up, and also that you use $x^ax^b=x^{a+b}$ and $x^{1/a}=\sqrt[a] x$ when dealing with exponentials.