$a$ is the product of $3$ and the square root of $2,$ $b$ is the product of $2$ and the square root of $3,$

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Mon Sep 07, 2020 7:21 am

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$a$ 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

Source: Beat The GMAT — Problem Solving | Mon Sep 07, 2020 7:21 am

swerve: Legendary Member; Posts: 2604; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

Re: $a$ is the product of $3$ and the square root of $2,$ $b$ is the product of $2$ and the square root of \(3

by swerve » Mon Sep 07, 2020 1:10 pm

Vincen wrote: ↑
Mon Sep 07, 2020 7:21 am
$a$ 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 $\dfrac{xy}{z}=\dfrac{(30+12\sqrt{6})\cdot (30-12\sqrt{6})}{36}=\dfrac{900+360\sqrt{6}-360\sqrt{6}-864}{36}=\dfrac{36}{36}=1$

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