Rationals or irrationals?

Determine whether the following numbers are rationals or irrationals.

a) $\displaystyle \mathcal{A}= \sqrt{33\cdot 5.\overline{15}+11\cdot 20.\overline{90}}$

b) $\displaystyle \mathcal{B}= \sqrt{\frac{\left | 33\cdot 5.\overline{15}-11\cdot 20.\overline{90} \right |}{10}}$

Solution

a) We are proving that it is rational:

$$\begin{align*}
\sqrt{33\cdot 5.\overline{15}+11\cdot 20.\overline{90}} &=\sqrt{33\cdot \left ( 5+ \frac{15}{99} \right )+11 \cdot \left ( 20+ \frac{90}{99} \right )} \\
 &= \sqrt{165+ 5+220+10}\\
 &= \sqrt{400}\\
 &=20
\end{align*}$$

b) We are proving that the number is irrational:

$$ \sqrt{\frac{\left | 33\cdot 5.\overline{15}-11\cdot 20.\overline{90} \right |}{10}} = \sqrt{\frac{\left | 170-230 \right |}{10}}= \sqrt{6}$$