\(a\) is the product of \(3\) and the square root of \(2,\) \(b\) is the product of \(2\) and the square root of \(3,\)

[Vincen]

\(a\) is the product of \(3\) and the square root of \(2,\) \(b\) is the product of \(2\) and the square root of \(3,\) and \(c\) is the product of \(2\) and the square root of \(6.\) If \(x\) is the square of the sum of \(a\) and \(b,\) \(y\) is the product of \(6\) and the difference of \(5\) and \(c,\) and \(z\) is the product of \(2\) squared and \(3\) squared, what is \(\dfrac{xy}{z} ?\)

A. \(1\)

B. \(30-12\sqrt6\)

C. \(36\)

D. \(30+12\sqrt6\)

E. \(64\)

Answer: A

Source: Princeton Review

[swerve]

\(a\) is the product of \(3\) and the square root of \(2,\) \(b\) is the product of \(2\) and the square root of \(3,\) and \(c\) is the product of \(2\) and the square root of \(6.\) If \(x\) is the square of the sum of \(a\) and \(b,\) \(y\) is the product of \(6\) and the difference of \(5\) and \(c,\) and \(z\) is the product of \(2\) squared and \(3\) squared, what is \(\dfrac{xy}{z} ?\)

A. \(1\)

B. \(30-12\sqrt6\)

C. \(36\)

D. \(30+12\sqrt6\)

E. \(64\)

Answer: A

Source: Princeton Review

If \(x\) is the square of the sum of \(a\) and \(b = (3\sqrt{2}+2\sqrt{3})^2=18+12\sqrt{6}+12=30+12\sqrt{6}\)

\(y\) is the product of \(6\) and the difference of \(5\) and \(c = 6\cdot (5-2\sqrt{6})=30-12\sqrt{6}\)

And \(z\) is the product of \(2\) squared and \(3\) squared, \(= 2^2\cdot 3^2=36\)

What is \(\dfrac{xy}{z}=\dfrac{(30+12\sqrt{6})\cdot (30-12\sqrt{6})}{36}=\dfrac{900+360\sqrt{6}-360\sqrt{6}-864}{36}=\dfrac{36}{36}=1\)