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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)
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BrentGMATPrepNow
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  22 Apr 2018, 05:15
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dave13 wrote:
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
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  08 May 2018, 08:29
(√x+√y)^2= x +y +2√xy

All needed info is present in two statements
C is answer

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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  27 Jul 2018, 00:23
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Bunuel wrote:
If x and y are positive integers, what is the value of \(\sqrt{x} + \sqrt{y}\)?

(1) x + y = 15
(2) \(\sqrt{xy}= 6\)


Kudos for a correct solution.


(1) 9+6=15
or, 12+3= 15
So, insufficient

(2) xy = 36
Which is : 4*9 = 36
12*3 = 36
So, there are two values of x and y.
insufficient.

Considering (1) and (2) x
([m]\sqrt{x} + \sqrt{y}[/m)^2= x+ y+2 [m]\sqrt{xy}
= 15+2.6
= 27
So, Ans is C.
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  21 May 2019, 09:22
I thought that that the GMAT requieres only unique solution. In this case √x + √y = ± √27 . Please help
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  16 Sep 2019, 03:52
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Bunuel wrote:
If x and y are positive integers, what is the value of \(\sqrt{x} + \sqrt{y}\)?

(1) x + y = 15
(2) \(\sqrt{xy}= 6\)


Kudos for a correct solution.


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
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  24 Apr 2020, 19:26
Bunuel wrote:
sushforgmat wrote:
GMATPrepNow wrote:

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

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 = 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:
Show Spoiler
C



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.

Apart from the original question:
Bunuel
If \((√x)^2=16\), shouldn't we get the value of √x=\(+/-\)√16 ?
Appreciating your help...
Thanks__
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  25 Apr 2020, 02:14
Expert Reply
Asad wrote:
Bunuel wrote:
sushforgmat wrote:
GMATPrepNow wrote:

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

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 = 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:
Show Spoiler
C



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.

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!!!)

\(x=16\)
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  29 Apr 2020, 22:24
Bunuel wrote:
Asad wrote:
Bunuel wrote:
sushforgmat wrote:

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.

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?
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  29 Apr 2020, 23:19
Expert Reply
Asad wrote:
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: \(\sqrt{x} \geq 0 \).
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  29 Apr 2020, 23:35
Bunuel wrote:
Asad wrote:
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: \(\sqrt{x} \geq 0 \).

Bunuel
So, what if the question prompt wouldn't say that x and y are positive? Will it effect this question if x and y are negative?
Thanks__
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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  25 Mar 2021, 03:21
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Solution:

St(1):- x + y = 15

Several values of x and y are possible and no unique value can be determined for x and y. (Insufficient)

St(2):- sqrt(xy) = 6

=> xy =36

=>x=y=6 or x=3 and y=12 or x=12 and y=3

Thus we cannot determine a unique answer to √x + √y .(Insufficient)

On combining both,

x + y = 15 & x=12 /y=3 or x=3 and y=12 (Since x=y=6 is not possible as x+ y =15)

Either ways, √x + √ y = √12 + √3 or √3 + √12 yielding the same value.

No other value is possible when combining both statements. (option c)

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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink] New post  06 May 2021, 22:25
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Bunuel wrote:
If x and y are positive integers, what is the value of \(\sqrt{x} + \sqrt{y}\)?

(1) x + y = 15
(2) \(\sqrt{xy}= 6\)


Kudos for a correct solution.


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Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2) [#permalink]
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