If x and y are positive integers less than 10

Best way to deal with such problems is try some values and POE:

Try x and y = 1, we get \(\sqrt{1} +\sqrt{1} = \sqrt{1 + 1}\)

2 = 1.4 --- Not true.

Try x = 2, y = 3, we get \(\sqrt{2} + \sqrt{3} = \sqrt{2 + 3}\)

3.1 = 2.2 ---- Not true.

This way you can check for few more values and you will realize (I) is not true.

Therefore, you can eliminate all options that have (I).

A,B,C, and E.

Hence D.

When I scan the statements, I see that ii and iii look easier, so I'll start with those.

ii) y√x = x√y
We can quickly see that, if x = y, then this statement is TRUE.
For example, if x = 1 and y = 1, we get 1√1 = 1√1, which is true.
So, statement ii COULD be true.
Check the answer choices......ELIMINATE A and C

iii) √x - √y = √(x - y)
Once again, this statement is TRUE when x = y.
For example, if x = 1 and y = 1, we get √1 - √1 = √(1 - 1), which is true.
So, statement iii COULD be true.
Check the answer choices......ELIMINATE B

i) √x + √y = √(x + y)
This is a tough one.
I can't think of any values for x and y that make this statement true, so now what?
Well, we might just conclude that statement i cannot be true.
However, we can do a bit of algebra to further convince ourselves.

Given: √x + √y = √(x + y)
Square both sides to get: (√x + √y)² = [√(x + y)]²
Expand to get: x + 2√(xy) + y = x + y
Subtract x and y from both sides to get: 2√(xy) = 0
This means that xy = 0, but xy cannot equal 0 since we're told that x and y are POSITIVE integers.
So, statement i cannot be true.

Answer: D

Cheers,
Brent

Solution:

If x = y, we see that Roman numerals II and III are true. However, Roman numeral I can never be true when x and y are positive integers. That is because, when we square both sides of the equation, we have:

LHS = (√x + √y)^2 = x + 2√x√y + y and RHS = (√(x + y))^2 = x + y

We see that, when each side is squared, the LHS is 2√x√y more than the RHS. They still can be equal if either x or y is 0. However, we are given that x and y are positive; therefore, 2√x√y can never be 0. In other words, (√x + √y)^2 > (√(x + y))^2 or √x + √y > √(x + y). So √x + √y = √(x + y) is a false statement when x and y are positive.

Answer: D

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