Question: Which of the following equations is true for all positive values of x and y?

1. \(\sqrt{x}+\sqrt{y}=\sqrt{x+y}\)
2. \(\sqrt{{x^4}{y^{16}}}=x^2y^4\)
3. \((x\sqrt{y})(y\sqrt{x})=x^2y^2\)
4. \(y\sqrt{x}+x\sqrt{y}=\sqrt{4xy^2}\)
5. \((x^y)(y^x)=(xy)^{2y}\)

“Which of the following equations is true for all positive values of x and y?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach Solution (1):

[A] \(\sqrt{x}+\sqrt{y}\neq \sqrt{x+y}\)

i.e., \(\sqrt4+\sqrt9\neq\sqrt13\)

[B] \(\sqrt{{x^4}{y^{16}}}=x^2y^4 \)

\(\sqrt{(x^2)(y^{16})}=\sqrt{(x^2)(y^8)^2}=(x)(y)^8\neq x^2 y^4\)

[C] \((x\sqrt{y})(y\sqrt{x})=x^2y^2\)

i.e., \(xy\sqrt{xy}-x^2y^2\)

[D] \(y\sqrt{x}+x\sqrt{y}=\sqrt{4xy^2}\)

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

\(\sqrt{4xy^2}=2y\sqrt{x}\)

Therefore, [D] is true for all positive values of x and y

[E] \((x^y)(y^x)=(xy)^{2y}\)

\((x^y)(y^y)\neq(xy)^{2y}\)

That is 4 * 9 \(\neq\) a big number

Correct option: D

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