Simplify $(1+\sqrt{3}) \cdot \sqrt{2-\sqrt{3}}$ Can someone help me simplify $(1+\sqrt{3})\times\sqrt{2-\sqrt{3}}$?
The end result is $\sqrt{2}$, however, I honestly do not know how to get there using my current skills.
I asked a teacher/tutor and he proposed setting the expression equal to X and working backwards, squaring both sides so that:
$$X^2 = (1 + \sqrt{3})^2 \cdot (2 - \sqrt{3}) =(4+2\sqrt{3})(2-\sqrt{3}) =8-2\sqrt{3}^2 = 2 \require{cancel}$$
$$\Rightarrow X = \sqrt{2}$$
My main question is:

*

*What are the steps to simplify this without setting equal to X?

*I tried watching a youtube video, but had no success - is difference of squares applicable here?

Thanks!

This is what the OP means by the 'setting equal to X' method.
Let $x = (1+\sqrt{3}) \cdot \sqrt{2-\sqrt{3}}$, and thus:
$$x^2 = (1 + \sqrt{3})^2 \cdot (2 - \sqrt{3}) =(4+2\sqrt{3})(2-\sqrt{3}) =8-2\sqrt{3}^2 = 2 \require{cancel}$$
$$\Rightarrow x = \cancel{-\sqrt{2}}, \sqrt{2}$$
The reason why we have crossed out $-\sqrt{2}$ is because $1 + \sqrt{3} > 1 > 0$, and $\sqrt{2 - \sqrt{3}} > 0$, thus $x = (1 + \sqrt{3})(\sqrt{2 - \sqrt{3}}) > 0$. Hence $x = \sqrt{2}$ is the only possible value.
