# How do you simplify -1*sqrt(2)*sqrt(2-sqrt(3)) ?

Oct 20, 2015

$1 - \sqrt{3}$

#### Explanation:

The rules for radicals say that we can multiply the terms of the same order together, so we can move the $2$ under the second radical term to get;

-1*sqrt(2(2-sqrt(3))

Multiplying the $2$ through, we get;

$- \sqrt{4 - 2 \sqrt{3}}$

A Google search on simplifying nested radicals turned up this page, which tells us that we can simplify this expression a little further using the rule;

$\sqrt{\left(x + y\right) - 2 \sqrt{x y}} = \sqrt{x} - \sqrt{y}$

Our expression is certainly the right form, but we need to check that we have reasonable $x$ and $y$ values to plug in. It turns out that;

$4 = \left(3 + 1\right)$
$3 = 3 \cdot 1$

Therefore, we can use $x = 3$ and $y = 1$ to simplify our expression.

$- \sqrt{\left(3 + 1\right) - 2 \sqrt{3 \cdot 1}}$

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

$= 1 - \sqrt{3}$

*Note: the page referenced above does not provide any proof for the equation mentioned, but if you square both sides, you can see that they are indeed equal.

${\left(\sqrt{\left(x + y\right) - 2 \sqrt{x y}}\right)}^{2} = {\left(\sqrt{x} - \sqrt{y}\right)}^{2}$

$\left(x + y\right) - 2 \sqrt{x y} = \left(\sqrt{x} - \sqrt{y}\right) \left(\sqrt{x} - \sqrt{y}\right)$

$x - 2 \sqrt{x y} + y = x - 2 \sqrt{x y} + y$