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Intern  Joined: 15 Feb 2012
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If root(xy) = xy what is the value of x + y?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 66% (01:39) correct 34% (01:31) wrong based on 324 sessions

### HideShow timer Statistics If $$\sqrt{xy} = xy$$ what is the value of x + y?

(1) x = -1/2
(2) y is not equal to zero

What i did was,

(XY)^1/2 = XY
XY =(XY)^2

so, I cancelled out XY and finally I got the below rephrased equation

XY=1.

BUT in the MGMAT explanation, I found that they are not cancelling out XY.

Below is their rephrased equation.

XY = (XY)^2

XY-(XY)^2=0

XY [1-(XY)] = 0

so, XY = 0 or XY = 1.

My question is why we are not cancelling out, and when we should use cancelling technique.
Math Expert V
Joined: 02 Sep 2009
Posts: 56307
Re: If root(xy) = xy what is the value of x + y?  [#permalink]

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If $$\sqrt{xy} = xy$$ what is the value of x + y?

$$\sqrt{xy} = xy$$ --> $$xy=x^2y^2$$ --> $$x^2y^2-xy=0$$ --> $$xy(xy-1)=0$$ --> either $$xy=0$$ or $$xy=1$$.

(1) x = -1/2 --> either $$-\frac{1}{2}*y=0$$ --> $$y=0$$ and $$x+y=-\frac{1}{2}$$ OR $$-\frac{1}{2}*y=1$$ --> $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Not sufficient.

(2) y is not equal to zero. Clearly not sufficient.

(1)+(2) Since from (2) $$y\neq{0}$$, then from (1) $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Sufficient.

As for your solution: you cannot divide by $$xy$$ since $$xy$$ could equal to zero and division by zero is not allowed.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero. So, if you divide (reduce) by $$xy$$ you assume, with no ground for it, that $$xy$$ does not equal to zero thus exclude a possible solution.

Hope it's clear.
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Re: If root(xy) = xy what is the value of x + y?  [#permalink]

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Responding to a pm:

"If $$\sqrt{XY} = XY$$ what is the value of x + y?

(1) x = -1/2
(2) y is not equal to zero

What i did was,

(XY)^1/2 = XY
XY =(XY)^2

so, I cancelled out XY and finally I got the below rephrased equation

XY=1.

BUT in the MGMAT explanation, I found that they are not cancelling out XY.

Below is their rephrased equation.

XY = (XY)^2

XY-(XY)^2=0

XY [1-(XY)] = 0

so, XY = 0 or XY = 1.

My question is why we are not cancelling out and when we should use cancelling technique."

For the time being, forget this question. Look at another one.

Which values of x satisfy this equation: $$x^2 = x$$
Let's say we cancel out x from both sides. What do we get? x = 1.
So we get that x can take the value 1.

But is your answer complete? I look at the equation and I say, 'x can also take the value 0.' Am I wrong? No.
x = 0 also satisfies your equation. So why didn't you get it using algebra? It is because you canceled x.
Let me treat this equation differently now.

$$x^2 = x$$
$$x^2 - x = 0$$
$$x * (x - 1) = 0$$
x = 0 OR (x - 1) = 0 i.e. x = 1

Now I get both the possible values that x can take. I do not lose a solution.

When you cancel off a variable from both sides of the equation, you lose a solution so you should not do that. You can cancel off constants of course.
Rule of thumb: Do not cancel off variables. Take them common. In some cases, it may not matter even if you do cancel off but either ways, your answer will not be incorrect of you don't cancel. On the other hand, sometimes, your solution could be incomplete if you do cancel and that's a problem.
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Intern  Joined: 12 Jul 2012
Posts: 21
Re: If root(xy) = xy what is the value of x + y?  [#permalink]

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Here is my solution

Take square of both sides:
(root(xy) )^2 = (xy)^2
xy = (xy)^2
With Option (1): x = -1/2
-1/2y = 1/4 * y^2
y = 4 * (-1/2)
y = -2
Hence, X+Y = -1/2 - 2 = -5/2.
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Joined: 02 Sep 2009
Posts: 56307
Re: If root(xy) = xy what is the value of x + y?  [#permalink]

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RamakantPareek wrote:
Here is my solution

Take square of both sides:
(root(xy) )^2 = (xy)^2
xy = (xy)^2
With Option (1): x = -1/2
-1/2y = 1/4 * y^2
y = 4 * (-1/2)
y = -2
Hence, X+Y = -1/2 - 2 = -5/2.

There is another value of $$y$$ apart -2 which satisfies $$-\frac{1}{2}*y=\frac{1}{4}*y^2$$, namely $$y=0$$.

Complete solution here: if-root-xy-xy-what-is-the-value-of-x-y-135646.html#p1103625
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If √(xy) = xy, what is the value of x + y?  [#permalink]

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If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.

If √(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

Statement 1: $$X=-1/2$$.... substitute in main equation

Scenario A: $$Y = -2/1$$ is a reciprocal
$$\sqrt{xy}=xy$$
LHS =$$\sqrt{-1/2*-2/1}= \sqrt{1}= 1$$
RHS =$$-1/2*-2/1= -1*-1 = 1$$
Therefore, LHS = RHS

Scenario B: $$Y = 0$$
LHS =$$\sqrt{-1/2*-0}= \sqrt{0}= 0$$
RHS =$$-1/2*0= 0$$
Therefore again, LHS = RHS

Hence, we have two possible solutions, therefore Statement 1 is insufficient

Statement 2 : $$Y$$ is not equal to zero
But, X could be equal to zero or be the reciprocal of Y, therefore, statement 2 would fall under the same scenarios as statement 1

Hence, we have two possible solutions, therefore Statement 2 is insufficient

Both Statement 1 & 2 together confirms that $$X & Y$$ both are not equal to zero, therefore, they have to be reciprocals and since statement gives us the value of $$X=-1/2$$, we can also the value of $$Y=-2/1$$ (Reciprocal of X) and thereafter we can find the value of $$X + Y = -1/2 + -2/1 = -5/2$$

Hence, both together are sufficient, therefore C is the correct Answer!

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Posts: 122
Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.

If √(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

I did something similar to DMMK, except for one thing:
I know $$\sqrt{n}$$ = $$n$$ only when $$n$$ = 0 or $$n$$ = 1. For all other values of $$n$$, the two would not be equal. Knowing this made it much simpler to plug in the statements. I just had to determine if from the statement(s) provided, can I rule out xy = 0 or xy=1.

Stament 1:
Knowing that x = -1/2, y could equal 0 or -2, and still make the premise true. Either value of y would make x + y a different value. Therefore, it is insufficient.

Statement 2:
Knowing that y is not equal to 0, x could equal zero or the inverse of y, and still make the premise true. Either value of x would make x + y a different value. Therefore, it is insufficient.

Now to evaluate both statements together.

The reason we rejected statement 1 by itself was because y could equal one of two possible values. Statement 2 eliminates one of those options. Therefore y must equal -2. And therefore both statements together are sufficient to answer "what is the value of x+y."
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.

If √(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

let xy=k, thus
k^(1/2) =k;
squaring both sides we have
k=k^2
k(k-1)=0
k=0 or 1
i.e. xy=0 or xy=1

st.1

for x=-1/2
both y=0 and y=-2 satisfy the possible value of xy i.e. 0 or 1
hence not sufficient

st.2

y is not equal to zero. clearly not sufficient.
as nothing is said about x, therefore x can take any value it can be zero, fraction, integer etc.

combining st.1 and st.2

we know that y cannot be equal to zero. thus y=-2 and x=-1/2
and x+y= -2-(1/2)=-5/2

hence sufficient.
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Re: If root(xy) = xy what is the value of x + y?  [#permalink]

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Why can't x and y both be -1? Thanks
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Posts: 56307
Re: If root(xy) = xy what is the value of x + y?  [#permalink]

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ElCorazon wrote:
Why can't x and y both be -1? Thanks

From the stem x = y = -1 is possible, in this case xy = 1 (check my solution above). But then the first statement says that x = -1/2, thus these values are no longer possible.

Hope it's clear.
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GMAT 1: 760 Q50 V44 Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.

If √(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

Given: √(xy) = xy
(xy)^2 - xy = 0
xy = 0 or 1 - (i)

Required: x + y = ?

Statement 1: x = -1/2
From the given equation (i), we can have two different values of y.
Hence two different values of x + y
INSUFFICIENT

Statement 2: y is not equal to 0
INSUFFICIENT

Combining Statement 1 and Statement 2: x = -1/2 and y is not equal to 0
From (i), xy cannot be 0 since both x and y are not = 0
Hence xy = 1
y = -2
x + y = $$-\frac{5}{2}$$
SUFFICIENT

Option C
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If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.

If √(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

The trick here is from $$\sqrt{xy}=xy$$ anyone could easily deduce that $$xy=1$$ or $$xy=0$$ and just consider one of these cases.

Here is my solution:
$$\sqrt{xy}=xy \implies \sqrt{xy}-xy=0 \implies \sqrt{xy}(\sqrt{xy}-1)=0$$
Hence we have $$xy=0$$ or $$xy=1$$.

(1) If $$x=-\frac{1}{2}$$, we need to consider two cases:

Case 1: If $$xy=0\implies y=0 \implies x+y = -\frac{1}{2}$$

Case 2: If $$xy=1 \implies y=-2 \implies x+y = -\frac{5}{2}$$

It's clear that (1) is insufficient.

(2) If $$y \neq 0$$, we still need to consider two cases:

Case 1: If $$xy=0\implies x=0 \implies x+y = y$$.

If $$y=1 \implies x+y = 1$$
If $$y=2 \implies x+y = 2$$

Hence, (2) is insufficient.

Now combine (1) and (2):
Since $$x \neq 0$$ and $$y \neq 0$$ hence $$xy \neq 0 \implies xy=1$$

Since $$x= -\frac{1}{2} \implies y = -2 \implies x+y = -\frac{5}{2}$$

Hence (1) & (2) are sufficient.
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.

If √(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

rewrite

xy= (xy)^2

St 1

Reciprocal property.

-1/2(-2) =1
[-1/2(-2)]^2 =1

insuff

St 2

Eliminating the possibility of 0 from Y still leaves the possibility of x being 0 or some other numbers which leads to several possibilities.

insuff

St 1 and St 2

Eliminates possibility of y=-2

C
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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Bunuel wrote:
If $$\sqrt{xy} = xy$$ what is the value of x + y?

$$\sqrt{xy} = xy$$ --> $$xy=x^2y^2$$ --> $$x^2y^2-xy=0$$ --> $$xy(xy-1)=0$$ --> either $$xy=0$$ or $$xy=1$$.

(1) x = -1/2 --> either $$-\frac{1}{2}*y=0$$ --> $$y=0$$ and $$x+y=-\frac{1}{2}$$ OR $$-\frac{1}{2}*y=1$$ --> $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Not sufficient.

(2) y is not equal to zero. Clearly not sufficient.

(1)+(2) Since from (2) $$y\neq{0}$$, then from (1) $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Sufficient.

As for your solution: you cannot divide by $$xy$$ since $$xy$$ could equal to zero and division by zero is not allowed.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero. So, if you divide (reduce) by $$xy$$ you assume, with no ground for it, that $$xy$$ does not equal to zero thus exclude a possible solution.

Hope it's clear.

Is it safe to say : x^2 = x only when x = 0 or 1
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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QZ wrote:
Bunuel wrote:
If $$\sqrt{xy} = xy$$ what is the value of x + y?

$$\sqrt{xy} = xy$$ --> $$xy=x^2y^2$$ --> $$x^2y^2-xy=0$$ --> $$xy(xy-1)=0$$ --> either $$xy=0$$ or $$xy=1$$.

(1) x = -1/2 --> either $$-\frac{1}{2}*y=0$$ --> $$y=0$$ and $$x+y=-\frac{1}{2}$$ OR $$-\frac{1}{2}*y=1$$ --> $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Not sufficient.

(2) y is not equal to zero. Clearly not sufficient.

(1)+(2) Since from (2) $$y\neq{0}$$, then from (1) $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Sufficient.

As for your solution: you cannot divide by $$xy$$ since $$xy$$ could equal to zero and division by zero is not allowed.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero. So, if you divide (reduce) by $$xy$$ you assume, with no ground for it, that $$xy$$ does not equal to zero thus exclude a possible solution.

Hope it's clear.

Is it safe to say : x^2 = x only when x = 0 or 1

Yes.

x^2 = x;

x^2 -x = 0;

x(x - 1) = 0;

x = 0 or x = 1.
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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Hi, why aren't we accounting for option where both x and y are equal to 1? then x+y is 2
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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krikre wrote:
Hi, why aren't we accounting for option where both x and y are equal to 1? then x+y is 2

Doesn't (1) say that x = -1/2? How it can be 1 then?
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Re: If √(xy) = xy, what is the value of x + y?  [#permalink]

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prakash111687 wrote:
If $$\sqrt{xy} = xy$$ what is the value of x + y?

(1) x = -1/2
(2) y is not equal to zero

What i did was,

(XY)^1/2 = XY
XY =(XY)^2

so, I cancelled out XY and finally I got the below rephrased equation

XY=1.
there are various cases that are possible as following

BUT in the MGMAT explanation, I found that they are not cancelling out XY.

Below is their rephrased equation.

XY = (XY)^2

XY-(XY)^2=0

XY [1-(XY)] = 0

so, XY = 0 or XY = 1.

My question is why we are not cancelling out, and when we should use cancelling technique.

IMO C
$$\sqrt{xy} = xy$$
1.x=0 ,y = anything
2.Y=0 , x=anything
3. both x and y 1 or -1 or x=1 and y=-1 or vise versa
4.x=1/z and y=z *any integer except 0
lets come to statement 1.
x=-1/2 y can be -2 or 0 insufficient.
2nd statement y not equal to 0 but don't know about x
combining both case with 0 is eliminated only one case left so C is the answer Re: If √(xy) = xy, what is the value of x + y?   [#permalink] 28 Feb 2018, 01:19
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