Problem Solving

After completing the GMAT Prep Now video sessions, I am happy to report that I scored a 640 on my practice test today. This is great, considering my first (and only so far) attempt at the GMAT (before the GMAT Prep Now videos) yielded a 430. There were two math problems in particular that didn't seem so difficult at first glance, yet I cannot seem to push through. This first one is difficult to convert to pure text, but I'll try:

Evaluate (2^((x+y)^2))/(2^((x-y)^2)) given that x*y = 1.

In other words, take x+y. Square that so you get (x+y) squared. Well, it's 2 to the power of all that. 2 to the power of (x+y) squared. That is the numerator. The demoninator is the same except x-y instead of x+y.

I expand the x+y squared and the x-y squared to get (x+y)(x+y) / (x-y)(x-y).
Then, I multiply to get x^2 + 2xy + y^2 / x^2 - 2xy + y^2.
Since x*y = 1, I get x^2 + 2 + y^2 / x^2 - 2 + y^2.

and that is where I'm stuck.
Any hints would be appreciated.

SoftwareDrone wrote:After completing the GMAT Prep Now video sessions, I am happy to report that I scored a 640 on my practice test today. This is great, considering my first (and only so far) attempt at the GMAT (before the GMAT Prep Now videos) yielded a 430. There were two math problems in particular that didn't seem so difficult at first glance, yet I cannot seem to push through. This first one is difficult to convert to pure text, but I'll try:

Evaluate (2^((x+y)^2))/(2^((x-y)^2)) given that x*y = 1.

In other words, take x+y. Square that so you get (x+y) squared. Well, it's 2 to the power of all that. 2 to the power of (x+y) squared. That is the numerator. The demoninator is the same except x-y instead of x+y.

I expand the x+y squared and the x-y squared to get (x+y)(x+y) / (x-y)(x-y).
Then, I multiply to get x^2 + 2xy + y^2 / x^2 - 2xy + y^2.
Since x*y = 1, I get x^2 + 2 + y^2 / x^2 - 2 + y^2.

and that is where I'm stuck.
Any hints would be appreciated.

Use exponent properties:
2^a/(2^b) = 2^(a-b)
Then:
2^(x+y)^2 / 2^(x-y)^2 = 2^( (x+y)^2 - (x-y)^2) = 2^( x^2 + y^2 + 2xy - ( x^2 + y^2 -2xy)) = 2^(4xy) = 2^4 = 16.

Oh my goodness how easy! I forgot about that property.
Thanks my friend!

SoftwareDrone wrote:After completing the GMAT Prep Now video sessions, I am happy to report that I scored a 640 on my practice test today. This is great, considering my first (and only so far) attempt at the GMAT (before the GMAT Prep Now videos) yielded a 430. There were two math problems in particular that didn't seem so difficult at first glance, yet I cannot seem to push through. This first one is difficult to convert to pure text, but I'll try:

Evaluate (2^((x+y)^2))/(2^((x-y)^2)) given that x*y = 1.

In other words, take x+y. Square that so you get (x+y) squared. Well, it's 2 to the power of all that. 2 to the power of (x+y) squared. That is the numerator. The demoninator is the same except x-y instead of x+y.

I expand the x+y squared and the x-y squared to get (x+y)(x+y) / (x-y)(x-y).
Then, I multiply to get x^2 + 2xy + y^2 / x^2 - 2xy + y^2.
Since x*y = 1, I get x^2 + 2 + y^2 / x^2 - 2 + y^2.

and that is where I'm stuck.
Any hints would be appreciated.

Let x=1 and y=1.
Then 2^(x+y)² / 2^(x-y)² = 2^(1+1)² / 2^(1-1)² = 2�/2� = 16.

Well that is a great idea! I didn't think I could assume that x = 1 and y = 1 just because x*y = 1, but, now that I think about it, I don't see why we couldn't do this.