Please help with two GMATPrep quant questions (exponents, ratios)

[alflauren]

I could really use some help on the solutions for these two GMATPrep questions:

1) If xy = 1, then what is the value of 2^(x+y)^2 / 2^(x-y)^2 ?

2) Kaye and Alberto have some stamps in a 5:3 ratio. Kaye gives Alberto 10 of her stamps, and the ratio becomes 7:5. How many more stamps does Kaye now have than Alberto?

Answers (highlight): #1) 16 #2) 40

Thanks!

[GMAT-HELP]

1) 2^(x^2+y^2+2xy)/2^(x^2+y^2 - 2xy) = 2^(4xy) = 2^4 = 16

2) Initial ratio = 5:3
=> After giving away 10 stamps, (5x-10)/(3x+10) = 7/5
=> 25x - 50 = 21x +70
=> x = 30
=> Kaye has 5x-10 = 140
=> Alberto has 3x+10 = 100
=> Kaye has 40 more stamps than Alberto.

[alflauren]

So the deal with this one:

2^(x^2+y^2+2xy)/2^(x^2+y^2 - 2xy)

is that the x^2 and y^2 cancel (and then the -2xy means that the 2^-2xy can be brought up to the top of the fraction)?

I guess I just haven't seen one like this before. Whenever I see addition/subtraction in a fraction, I am trained to think that you can never cancel. I guess that's not the case if the addition is in an exponent.

[longatum]

A^2-B^2 = (A-B)(A+B) (1)
A^n/A^m = A^(n-m) (2)

--> 2^((x+y)^2)/2^((x-y)^2) = 2^[(x+y)^2-(x-y)^2] (according to (2)) = 2^[(x+y+x-y)(x+y-x+y)] (according to (1)) = 2^(4xy) =16

OR you can do it like GMAT-HELp and utilize only (2) to bring the denominator up and get the same result.