Question

An operation \( \$ \) is defined as follows. For any two positive integers \(x\) and \(y\),
\[ x \$ y = \sqrt{ \sqrt{\dfrac{x}{y}} + \sqrt{\dfrac{y}{x}} } \]
Which of the following is an integer?

• A. \(4 \$ 9\)
• B. \(4 \$ 16\)
• C. \(4 \$ 4\)
• D. None of the above

Hint:

Plug the given pairs into the formula one at a time and simplify the nested square roots fully before deciding if the result is a whole number.

Answer 1

The Correct Option is D

Step 1: Rewrite the rule in one variable.
Let $r = x/y$. Then $y/x = 1/r$, so the operation becomes a function of just $r$:
\[ x \$ y = \sqrt{ \sqrt{r} + \dfrac{1}{\sqrt{r}} } \]
This is a cleaner way to test all three cases, since we only need the ratio $x/y$ for each pair, not the two numbers separately.

Step 2: Find $r$ for each option.
For $4 \$ 9$, $r = 4/9$.
For $4 \$ 16$, $r = 4/16 = 1/4$.
For $4 \$ 4$, $r = 4/4 = 1$.

Step 3: Evaluate the function at $r = 4/9$.
$\sqrt{r} = 2/3$ and $1/\sqrt{r} = 3/2$.
Sum $= 2/3 + 3/2 = 13/6$, which is not a perfect square of a rational number, since 13 is prime and does not divide 6. So $\sqrt{13/6}$ is irrational, and $4 \$ 9$ fails.

Step 4: Evaluate the function at $r = 1/4$.
$\sqrt{r} = 1/2$ and $1/\sqrt{r} = 2$.
Sum $= 1/2 + 2 = 5/2$. Again 5 and 2 share no square factor, so $\sqrt{5/2}$ is irrational. $4 \$ 16$ fails too.

Step 5: Evaluate the function at $r = 1$.
$\sqrt{r} = 1$ and $1/\sqrt{r} = 1$.
Sum $= 1 + 1 = 2$, and $\sqrt{2}$ is a classic irrational number. $4 \$ 4$ fails as well.

Step 6: Conclude.
All three specific cases give an irrational square root, never a whole number. So none of the given options works, and the answer has to be none of the above.
\[ \boxed{\text{None of the above}} \]