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Math Expert V
Joined: 02 Sep 2009
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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Question Stats: 76% (01:16) correct 24% (01:24) wrong based on 1484 sessions

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

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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) Gives you various single values for x and y. Therefore clearly insufficient.
2) If $$\sqrt{xy}= 6$$, then xy = 36 which can be built with 3*12 or 6*6 ... insufficient.

1+2) Here we know, x+y = 15 and xy = 36, hence x, or y are splitted up as 12 and 3. It does actually not matter if x is 3 or 12 or y is 3 or 12. The sum of $$\sqrt{x} + \sqrt{y}$$ will be the same.

##### General Discussion
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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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(sqrt(x) + sqrt(y))^2 = x + y + 2(sqrt(xy))

Statement 1: Not Sufficient
Statement 2: Not Sufficient

Combining St1 and St2 we have the values for (x + y) and sqrt(xy) - Sufficient

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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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4
((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
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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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6
2
To find the value of $$\sqrt{x} + \sqrt{y}$$ we need to know to have a value for x and a value for y.

Statement 1 : INSUFFICIENT
x + y = 15
We have different possible values for x and y:
x= 7 and y= 8
x= 9 and y= 6
x= 12 and y=3
All of these would yield different values for $$\sqrt{x} + \sqrt{y}$$. Since we can't find a unique value, the statement is not sufficient.

Statement 2 : INSUFFICIENT
If $$\sqrt{xy}=6$$ then $$(\sqrt{xy})^2=6^2$$ and $$xy=$$36.
Again, there are multiple values of x and y for which $$xy=36$$:
x=36 and y=1
x=6 and y=6
Since we can't find a unique value, the statement is not sufficient.

(1) + (2) = SUFFICIENT

We know that x+y = 15 and that xy=36, because x and y are positive integers we know that x=12 and y=3 OR x=3 and y=12 either way we will be able to calculate the value of $$\sqrt{x} + \sqrt{y}$$ because it will not change the result.

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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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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.
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Joined: 02 Sep 2009
Posts: 60492
Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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1
malavika1 wrote:
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.
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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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Bunuel wrote:
malavika1 wrote:
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.

Sorry to post little unrelated post; where should we consider mode in GMAT. As I remember, in one of your post, you mentioned -- sqrt(x^2)=|x|
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Joined: 02 Sep 2009
Posts: 60492
Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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AR15J wrote:
Bunuel wrote:
malavika1 wrote:
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.

Sorry to post little unrelated post; where should we consider mode in GMAT. As I remember, in one of your post, you mentioned -- sqrt(x^2)=|x|

Not following you... What is your question?

P.S. Yes, $$\sqrt{x^2}=|x|$$.
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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Top Contributor
1
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.

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: http://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

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Originally posted by GMATPrepNow on 01 Aug 2017, 12:29.
Last edited by GMATPrepNow on 07 Sep 2018, 16:14, edited 1 time in total.
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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1
1
Bunuel wrote:
If x and y are positive integers, what is the value of $$\sqrt{x} + \sqrt{y}$$?

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

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.

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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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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.

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 $$x$$.

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.
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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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GMATPrepNow wrote:
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.

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

But, isn't √x + √y = ± √27 which would not result in a single solution for the question?
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Joined: 02 Sep 2009
Posts: 60492
Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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1
sushforgmat wrote:
GMATPrepNow wrote:
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.

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

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.
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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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Bunuel wrote:
sushforgmat wrote:
GMATPrepNow 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.

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

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.

With multiple books and reading a lot of content, I think I missed the basic point that you mentioned.
Thanks, Bunuel.
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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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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?

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Joined: 02 Sep 2009
Posts: 60492
Re: If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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1
bbg wrote:
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?

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, $$\sqrt{2 + 2} \neq \sqrt{2}+\sqrt{2}$$.
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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Bunuel chetan2u niks18 amanvermagmat

Quote:
If x and y are positive integers, what is the value of $$\sqrt{x} + \sqrt{y}$$?

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

I think we over complicated the solution.
Can we not simply approach q as : $$(a+b)^2$$ = $$a^2$$+ $$b^2$$ + 2 ab

Substitute $$\sqrt{a}$$ for a and $$\sqrt{b}$$ for b
only both statements together help in completing equation.

Is this method correct? _________________
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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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Bunuel chetan2u niks18 amanvermagmat

Quote:
If x and y are positive integers, what is the value of $$\sqrt{x} + \sqrt{y}$$?

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

I think we over complicated the solution.
Can we not simply approach q as : $$(a+b)^2$$ = $$a^2$$+ $$b^2$$ + 2 ab

Substitute $$\sqrt{a}$$ for a and $$\sqrt{b}$$ for b
only both statements together help in completing equation.

Is this method correct? Yes it is perfectly fine and simpler. Square both sides and then take the square root of the resulting value and finally discard the negative value as x and y are positive

Posted from my mobile device
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If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)  [#permalink]

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GMATPrepNow wrote:
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.

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: http://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 = 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

RELATED VIDEO

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  If x and y are positive integers, what is the value of x^(1/2)+y^(1/2)   [#permalink] 22 Apr 2018, 05:33

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