Question

Help!
which sum is rational?

A.)
$\pi + 18$
B.)
$\sqrt{25 + 1.75}$
C.
$\sqrt{3 + 5.5}$
D.)
$\pi + \sqrt{2}$

Answer 1

$\pi+18-irrational\ because\ \pi\ is\ irrational\\\\\sqrt{25+1.75}=\sqrt{26.75}=\sqrt{\dfrac{2675}{100}}=\dfrac{\sqrt{2675}}{\sqrt{100}}=\dfrac{\sqrt{25\cdot107}}{10}\\\\=\dfrac{\sqrt{25}\cdot\sqrt{107}}{10}=\dfrac{5\sqrt{107}}{10}=\dfrac{\sqrt{107}}{2}-irrational\\\\\sqrt{3+5.5}=\sqrt{8.5}=\sqrt{\dfrac{85}{10}}=\sqrt{\dfrac{850}{100}}=\dfrac{\sqrt{850}}{\sqrt{100}}=\dfrac{\sqrt{25\cdot34}}{10}\\\\=\dfrac{\sqrt{25}\cdot\sqrt{34}}{10}=\dfrac{5\sqrt{34}}{10}=\dfrac{\sqrt{34}}{2}-irrational$

$\pi+\sqrt2-irrational$