Which of the following numbers does not have a square root in the form \(x + y\sqrt{2}\), where \(x\) and \(y\) are positive integers?
- \(17+12\sqrt{2}\)
- \(22+12\sqrt{2}\)
- \(38+12\sqrt{2}\)
- \(54+12\sqrt{2}\)
- \(73+12\sqrt{2}\)
Let’s suppose \(x + y\sqrt{2}\) is the square root of \(z\).
So \(z^2 = (x + y\sqrt{2})^2 = (x^2+2y^2) + (2xy)\sqrt{2}\).
We need \(2xy = 12\), and so the only possible pairs \((x,y)\) for this to be true are \((1,6), (6,1), (2,3)\) and \((3,2)\).
\((x,y) = (1,6)\implies z^2 = 73+12\sqrt{2}\)
\((x,y) = (6,1)\implies z^2 = 38+12\sqrt{2}\)
\((x,y) = (2,3)\implies z^2 = 22+12\sqrt{2}\)
\((x,y) = (3,2)\implies z^2 = 17+12\sqrt{2}\)
Thus the odd one out is \(54+12\sqrt{2}\), and the answer is (D).
