Review question

# Which of these numbers does not have a square root of the form $x + y\sqrt{2}$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6256

## Solution

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?

1. $17+12\sqrt{2}$
2. $22+12\sqrt{2}$
3. $38+12\sqrt{2}$
4. $54+12\sqrt{2}$
5. $73+12\sqrt{2}$

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).