Question

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

Answer 1

`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(4-2sqrt(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((x+y)-2sqrt(xy))=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=(3+1)`
`3=3*1`

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

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

`= -(sqrt(3)-sqrt(1))`

`=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.

`(sqrt((x+y)-2sqrt(xy)))^2=(sqrtx - sqrty)^2`

`(x+y)-2sqrt(xy)=(sqrtx-sqrty)(sqrtx-sqrty)`

`x-2sqrt(xy) +y = x-2sqrt(xy) + y`