If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)

((x)^(1/2)+(y)^(1/2))^2 = x + y + 2 *(xy)^(1/2)

1. x+y= 15
Not sufficient

2.
(xy)^(1/2) = 6
Not sufficient

Combining 1 and 2, we get
x + y + 2 *(xy)^(1/2)= 15 + 2*6=27
=> ((x)^(1/2)+(y)^(1/2))^2 = 27
=> (x)^(1/2)+(y)^(1/2) =3*((3)^(1/2)

Sufficient
Answer C

Whenever we are given that for example, root(n) = something,

Can we always pretty much blindly conclude that n = (something)^2?

Or is there something we have to watch out for.

If we are given that say \(\sqrt{x}=y\), then we can square and get x = y^2.

Target question: What is the value of √x + √y?

Statement 1: x + y = 15
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 14 and y = 1, in which case √x + √y = √14 + √1 = √14 + 1
Case b: x = 9 and y = 6, in which case √x + √y = √9 + √6 = 3 + √6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: √(xy) = 6
In other words xy = 36
This statement doesn't FEEL sufficient either, so I'll TEST some values.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = 36, in which case √x + √y = √1 + √36 = 1 + 6 = 7
Case b: x = 4 and y = 9, in which case √x + √y = √4 + √9 = 2 + 3 = 5
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x + y = 15
Statement 2 tells us that √(xy) = 6
Recognize that (√x + √y)² = x + 2√(xy) + y
Rearrange to get: (√x + √y)² = x + y + 2√(xy)
We get: (√x + √y)² = 15 + 2(6)
Evaluate: (√x + √y)² = 27
So, √x + √y = √27
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

We need to determine the value of √x + √y.

Statement One Alone:

x + y = 15

If x = 1 and y = 14, then √x + √y = 1 + √14. However, if x = 4 and y = 11, then √x + √y = 2 + √11. We see that we don’t have enough information to determine a unique value of √x + √y.

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

√(xy) = 6

If x = 6 and y = 6, then √x + √y = 2√6. However, if x = 4 and y = 9, then √x + √y = 5. We see that we don’t have enough information to determine a unique value of √x + √y.

Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Notice that (√x + √y)^2 = x + y + 2√(xy). From the two statements, we are given that x + y = 15 and √(xy) = 6, and thus (√x + √y)^2 = 15 + 2(6) = 27. Now, if we take the square root of both sides of the equation (√x + √y)^2 = 27, we have √x + √y = √27 = 3√3.

Answer: C

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.


Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer and so we should consider 1) & 2) first.

\((\sqrt{x} + \sqrt{y})^2 = x + 2\sqrt{xy} + y = x + y + 2\sqrt{xy} = 15 + 2 \cdot 6 = 15 + 12 = 27\).
Both conditions 1) & 2) are sufficient.

Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT 4(A).

Condition 1)
Since \(y = 15 - x\), we have \(\sqrt{x} + \sqrt{y} = \sqrt{x} + \sqrt{15 - x}\).
However, the condition 1) is not sufficient since we don't know \(x\).

Condition 1)
Since \(xy = 36\) and \(y = \frac{36}{x}\), we have \(\sqrt{x} + \sqrt{y} = \sqrt{x} + \sqrt{\frac{36}{x}}\).
However, the condition 1) is not sufficient since we don't know \(x\).

Therefore, C is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

But, isn't √x + √y = ± √27 which would not result in a single solution for the question?

The square root of a number (generally even root of a number) is non-negative: 0 or positive. \(\sqrt[even]{nonnegative \ number}\geq 0\). Thus, \(\sqrt{x} + \sqrt{y} = {nonnegative \ number} + {nonnegative \ number}= {nonnegative \ number}\), so it cannot equal to a negative number.

I thought (1) alone was enough since if
x+y=15
I could do the squared root of each term
√x + √y = ± √15
Am I breaking some math rules?

Thank you in advance

If you take the square root from x + y = 15, you'll get \(\sqrt{x + y} = \sqrt{15}\), which is NOT the same as \(\sqrt{x}+\sqrt{y} = \sqrt{15}\). You see, generally, \(\sqrt{x + y} \neq \sqrt{x}+\sqrt{y}\). For example, \(\sqrt{2 + 2} \neq \sqrt{2}+\sqrt{2}\).

hello there GMATPrepNow

can you please explain how you combine both statements to get answer

Statement 1 tells us that x + y = 15
Statement 2 tells us that √(xy) = 6

and then you say Recognize that (√x + √y)² = x + 2√(xy) + y

I dont recognize this pattern in either of the statements how can I recognize it looking at both statements where did you get this formula ?

I thought I should do square both sides of √(xy) = 6 so I am getting xy = 36 and also have x + y = 15 so I do something like this

x + y = 15 ---> x = 15-y and plug in here xy = 36

thank you in advance for taking time to explain and have a great gmat weekend

It's not easy to see that (√x + √y)² = x + 2√(xy) + y, however when we see that we have information about x and y and we need to find the value of √x and √y and y, we might start looking for possible relationships that might help us. One relationship is that (√x)² = x and that (√y)² =y
From there, it's a matter of exploring what (√x + √y)² might look like.
We can use the FOIL method to expand (√x + √y)²
We get: (√x + √y)² = (√x + √y)(√x + √y) = x + √xy + √xy +y = x + 2√xy + y
At this point, we can see that our exploration has paid off.

Keep in mind that solving math questions involves little explorations like this. Sometimes those explorations take us closer to the solution; sometimes they take us nowhere, in which case we need to try something else.

Cheers,
Brent

If x and y are positive integers, what is the value of \(\sqrt{x} + \sqrt{y}\)?

\(\sqrt{x} + \sqrt{y} =\sqrt{(x+y+2\sqrt{xy})}\)

(1) x + y = 15
NOT SUFFICIENT

(2) \(\sqrt{xy}= 6\)
NOT SUFFICIENT

(1) + (2)
(1) x + y = 15
(2) \(\sqrt{xy}= 6\)
\(\sqrt{x} + \sqrt{y} =\sqrt{(x+y+2\sqrt{xy})} = \sqrt{(15+2*6)} = \sqrt{27}\)
Since x & y are positive integers, \(\sqrt{x} + \sqrt{y}\) is positive

SUFFICIENT

IMO C

Apart from the original question:
Bunuel
If \((√x)^2=16\), shouldn't we get the value of √x=\(+/-\)√16 ?
Appreciating your help...
Thanks__

\((\sqrt{x})^2=16\);

\(\sqrt{x}=4\). ( \(\sqrt{x}\) CANNOT BE NEGATIVE!!!)

\(x=16\)

Bunuel

Yes, we know that we can't put any negative value inside of root-it gives imaginary number. So, WHY GMAC forces us to consider x and y is positive (not negative), at least for this question?

Two different points regarding even roots:

1. An even root to be defined (on the gmat) the expression under the root must be positive or 0: \(\sqrt{x}\) to be defined x must be positive or 0.

2. An even root CANNOT give negative result: \(\sqrt{x} \geq 0 \).