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605-655 (Medium)|   Algebra|      
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Bunuel
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If (xy)^)(1/2) = xy , what is the value of x + y? 30 Jan 2023, 00:05
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puneetfitness
Bunuel
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.

Answer: C.


Hi Bunuel when we have xy=(xy)^2 why we cannot assume that (xy)^2/xy=1 why should we do (xy)^2 -xy=o

My question is how do we know when the power on two side of equality should cancel out to simplify and when should we perform subtract as done above

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You cannot reduce xy=(xy)^2 by xy because xy can be 0 and we cannot divide by 0. By doing so you loose a root, namely xy = 0.

Never reduce equation by a variable (or expression with a variable), if you are not certain that the variable (or expression with the variable) doesn't equal to zero. We cannot divide by zero.
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