# How do you simplify square root of 30 - square root of 3?

Jul 3, 2018

Both $\sqrt{30} - \sqrt{3}$ and $\sqrt{30 - \sqrt{3}}$ are in simplest form already.

#### Explanation:

It is not clear from the question whether you intend:

$\sqrt{30} - \sqrt{3}$

or:

$\sqrt{30 - \sqrt{3}}$

Note that:

$30 = 2 \cdot 3 \cdot 5$

has no square factors, so cannot be simplified.

However, note that one of its factors is $3$. So it is possible to split $\sqrt{30}$ into a product of square roots that we can group as follows:

$\sqrt{30} - \sqrt{3} = \sqrt{10} \cdot \sqrt{3} - 1 \cdot \sqrt{3} = \left(\sqrt{10} - 1\right) \sqrt{3}$

I am not sure that you would call that a simplification, but it might be a useful alternative form.

In the case of $\sqrt{30 - \sqrt{3}}$ we might try looking for an expression of the form $a + b \sqrt{3}$ whose square is $30 - \sqrt{3}$, but this does not lead to a simplification...

${\left(a + b \sqrt{3}\right)}^{2} = \left({a}^{2} + 3 {b}^{2}\right) + \left(2 a b\right) \sqrt{3}$

So:

$\left\{\begin{matrix}{a}^{2} + 3 {b}^{2} = 30 \\ 2 a b = - 1\end{matrix}\right.$

Putting $b = - \frac{1}{2 a}$ the first equation becomes:

${a}^{2} + \frac{3}{4 {a}^{2}} = 30$

and hence:

${a}^{4} - 30 {a}^{2} + \frac{3}{4} = 0$

and hence:

$4 {a}^{4} - 120 {a}^{2} + 3 = 0$

from which we find:

${a}^{2} = 15 \pm \frac{\sqrt{897}}{2}$

So there are no nice rational or simple irrational values of $a , b$ such that ${\left(a + b \sqrt{3}\right)}^{2} = \sqrt{30 - \sqrt{3}}$