A PSAT Sample Test Problem :: Help, cannot understand it!

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Dark Knight 496

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Oct 8, 2005
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This is a problem from a PSAT sample test that I have just completed.
There was one problem on it that I didn't understand... Here it is:

If 8*SQRT(8) = X*SQRT(Y), and x and t are different positive integers, what is one possible value of X+Y?

According to the answer key, the possible answers are: 18, 36, 130, 513
However, I have no idea how to get those answers.
Any help is appreciated; SQRT means sq. root.
Thanks in advance :)
 
18 is one possible answer because 8sqrt(8)=16sqrt(2), 2+16=18.

Using the laws of radicals:

8sqrt(8)=8sqrt(2)sqrt(4)=8sqrt(2)*2=16sqrt(2)
 
Re: A PSAT Sample Test Problem :: Help, cannot understand it

Dark Knight 496 said:
If 8*SQRT(8) = X*SQRT(Y), and x and t are different positive integers, what is one possible value of X+Y?
Click to expand...
How does "t" relate to the question? Or is that supposed to be a "Y"? And is "X" the same as "x", or is there further information relating these four variables?

Thank you.

Eliz.
 
Yes, sorry, t is a typo and should be a y.
In my post, X = x and Y = y...
Thanks.

Fixed version:

============================================
If 8*sqrt(8) = x*sqrt(y), and x and y are different positive integers, what is one possible value of x+y?
============================================
 
We have:

. . . . .8sqrt(8) = xsqrt(y), with x not equal to y

Okay, we can simplify this right now, by noting that 8 = 2<sup>3</sup>, so sqrt(8) = 2sqrt(2), and 8sqrt(8) = 16sqrt(2). Right there, you've got "x = 16, y = 2" as a possible solution.

You could, of course, go the other way, by rearranging in a non-simplifying direction. Since 8 = (2)(4) and since 4 = sqrt(16), then 8 = 2sqrt(16). And since 8 = (2)(4) and since 2 = sqrt(4), then 8 = 4sqrt(4). And since 8 = sqrt(64), 8sqrt(8) = sqrt(64)sqrt(8). And so forth.

Fiddle around with that, and see if you can get their other options.

Eliz.
 
An Unco approach:

x = 8sqrt(8)/sqrt(y)
x = 16sqrt(2)/sqrt(y)

Let y = 2*t^2 for t is a positive integer

x = 16sqrt(2)/sqrt(2t^2)
x = 16sqrt(2)/[t.sqrt(2)]
x = 16/t

x is a positive integer if t is a positive factor of 16
That is, if t = 1, 2, 4, 8, 16

t=1: x=16/1=16, y=2*1^1=2 --> x+y=18
t=2: x=16/2=8, y=2*2^2=8... but we are told x is not equal to y
t=4: x=16/4=4, y=2*4^2=32 --> etc.
 
stapel,

Darn.. That solution is so simple. How come I did not see that 8 = 2*sqrt(2)? :p Thanks.

Unco,

Nice approach. Although it takes time to write it all out, but at least I can see how they got their answers.
Thank you!
 
Dark Knight 496 said:
How come I did not see that 8 = 2*sqrt(2)?
Click to expand...
It's an easy mistake to make. Sometimes we get so centered on finding "the formula" or "the method" that we're "supposed" to use, that we neglet the very powerful technique of "just fiddling with it, and seeing what happens."

Never be afraid to mess around with the math, to see if something useful falls out. :wink: :D

Eliz.
 
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