Question

Show that 2 + 3 root 2 the whole divided by 7 is not a rational

Answer 1

Dear Student,
$$\begin{gathered} \frac{2+3\sqrt{2}}{7} \\ \frac{2}{7}+\frac{3}{7}\times \sqrt{2} \end{gathered}$$

and
Assume $\sqrt{2}$ is rational, i.e. it can be expressed as a rational fraction of the form $\frac{\mathrm{b}}{\mathrm{a}}$, where a and b are two relatively prime integers.
$$\begin{gathered} 2=\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2} \\ \Rightarrow 2{\mathrm{b}}^2={\mathrm{a}}^2 \end{gathered}$$

Since $2{\mathrm{a}}^2$ is even, ${\mathrm{b}}^2$ must be even, and therefore b is also even. Now let
$$\begin{gathered} \mathrm{b}=2\mathrm{c} \\ \Rightarrow 4{\mathrm{c}}^2=2{\mathrm{a}}^2 \\ \Rightarrow {\mathrm{a}}^2=2{\mathrm{c}}^2 \end{gathered}$$
Which means ${\mathrm{a}}^2$ is even, and so is a.
However, two even numbers cannot be relatively prime. So, $\sqrt{2}$ cannot be expressed as a rational fraction, Therefore it is irrational.
Now, in the given question,
$\frac{3}{7}\times \sqrt{2}$ will be irrational as irrational quantity is multiplied by a rational quantity and
this term added to $\frac{2}{7}$ will give an irrational number as a result.
Therefore, $\frac{2+3\sqrt{2}}{7}$ is not a rational number.
Regards