Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 2 Exercises 2.7

Our main tool is Corollary 2.3.

Solution:

### Part a

If $\sqrt{4+2\sqrt 3}-\sqrt 3$ is rational. Let us assume it to be $x$, we would like to determine $x$. Then we must have

$$

\sqrt{4+2\sqrt 3}=x+\sqrt{3}.

$$ Taking square of both sides, we obtain

\begin{equation}\label{2-7}

4+2\sqrt{3}=x^2+2x\sqrt 3+3.

\end{equation} Since $x$ is rational, we should expect that $2\sqrt{3}=2x\sqrt 3$ which is true only when $x=1$. Moreover, \eqref{2-7} is true if $x=1$. We have

$$

4+2\sqrt{3}=1+2\sqrt 3+3=(1+\sqrt 3)^2.

$$

Hence $\sqrt{4+2\sqrt 3}=1+\sqrt 3$ and $\sqrt{4+2\sqrt 3}-\sqrt 3=1$ is rational.

### Part b

This works very similar as Part a. We only give an outline. Note that

$$

(2+\sqrt 2)^2=4+4\sqrt 2+2=6+4\sqrt 2.

$$ Therefore

$$

\sqrt{6+4\sqrt 2}-\sqrt 2=(2+\sqrt 2)-\sqrt 2=2

$$ is rational.