# How do you simplify: 2/square root of 10?

Jul 21, 2015

$\frac{\sqrt{10}}{5}$

#### Explanation:

The typical way this is taught is to "rationalize the denominator"

$\frac{2}{\sqrt{10}} = \frac{2}{\sqrt{10}} \cdot \setminus \frac{\sqrt{10}}{\setminus} \sqrt{10} = \frac{2 \setminus \sqrt{10}}{10} = \setminus \frac{\sqrt{10}}{5}$

In general, if you want to write an expression of the form $\frac{a}{\sqrt{b}}$ in the form $\frac{c \sqrt{d}}{e}$, you need to multiply the first expression by $\frac{\sqrt{b}}{\sqrt{b}}$. In this case, that gives:

$\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a \sqrt{b}}{b}$ (so $c = a$, $b = d = e$)

At this point, it's possible that $a$ and $b$ have common factors that can cancel to make the fraction even simpler.

Why is this done? Probably two main reasons:

1) It gives these kinds of numbers a standard form that makes it easier for teachers to check the answers.

2) It's a useful skill for higher-level math, such as in calculus, when certain kinds of "limits" are evaluated using this technique.

You should not make the mistake of thinking that square roots in denominators are somehow "bad" or "wrong". It's not like you are dividing by zero or something.