phoenixio wrote:
x and y are positive integers such that x < y. If \(6\sqrt{6}= x\sqrt{y}\), then xy could equal
A)36
B)48
C)54
D)96
E)108
Here's another approach:
GIVEN: (6)(√6) = (x)(√y)
Rewrite 6 as √36 to get: (√36)(√6) = (x)(√y)
Rewrite x as √(x²) to get: (√36)(√6) = √(x²)(√y)
Simplify both sides to get: √216 = √(x²y)
From this, we can conclude that 216 = x²y
If x and y are positive integers, what are some possible values of x and y?
To help us with this, let's find the prime factorization of 216
216 = (2)(2)(2)(3)(3)(3)
So, we can write:
(2)(2)(2)(3)(3)(3) = x²yNow it's a matter of looking for possible values of x and y that meet the above condition.
Here's one possibility: 216 = (9)(24) = (3²)(24)
In other words, x = 3 and y = 24
In this case, xy = (3)(24) = 72. 72 is NOT among the answer choices.
KEEP LOOKING!
Here's another possibility: 216 = (4)(54) = (2²)(54)
In other words, x = 2 and y = 54
In this case, xy = (2)(54) = 108.
108 IS among the answer choices!
Answer: E
Cheers,
Brent