If 2^(x + y) = 4^8, then

jjjinapinch wrote:If 2^(x + y) = 4^8, then what is the value of y?
(1) x² = 81
(2) x - y = 2

Official Guide question
Answer: B

Target question: What is the value of y?

Given: 2^(x + y) = 4^8
Rewrite right side with base of 2.
So, first replace 4 with 2² to get: 2^(x + y) = (2²)^8
Simplify right side: 2^(x + y) = 2^16
We can conclude that x + y = 16

Statement 1: x² = 81
This tells us that EITHER x = 9 OR x = -9
Let's test each case.
Case a: If x = 9, then we can take x + y = 16 and replace x with 9 to get: 9 + y = 16. When we solve this equation for y, we get y = 7
Case b: If x = -9, then we can take x + y = 16 and replace x with -9 to get: -9 + y = 16. When we solve this equation for y, we get y = 25
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x - y = 2
Since we also know that x + y = 16 , we now have a system of 2 variables with equations, which we COULD solve for y (but we won't do because we don't want to waste valuable time).
Since we COULD answer the target question with certainty, statement 2 is SUFFICIENT

Aside: For "fun" let's solve the following system:
x + y = 16
x - y = 2

If we SUBTRACT the bottom equation from the top equation, we get: 2y = 14, which means y = 7
Voila!

Answer: B

Cheers,
Brent

Hi All,

We're told that 2^(X + Y) = 4^8. We're asked for the value of Y. This question is based heavily on how you recognize patterns and organize your information. To start, we can rewrite the 'right side' of the equation to put it in a 'power of 2.' Since 4 = 2^2, the value 4^8 can be rewritten as (2^2)^8 = 2^16. Thus...

2^(X + Y) = 2^16.... meaning that (X+Y) = 16.

1) X² = 81

With the information in Fact 1, we know that X could be +9 or -9
IF....
X = +9, then Y=7
X = -9, then Y=25
Fact 1 is INSUFFICIENT

2) (X - Y) = 2

Fact 2 gives us a second unique equation using the variables X and Y, so we have a 'system' of equations and we CAN solve for the exact values of X and Y. We don't actually have to do that work, but if we did, we'd find that X=9 and Y=7.
Fact 2 is SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

jjjinapinch wrote:If 2^(x + y) = 4^8, then what is the value of y?
(1) x^2 = 81
(2) x - y = 2

We can simplify the given equation to:

2^(x + y) = 2^16

x + y = 16

We need to determine the value of y.

Statement One Alone:

x^2 = 81

Since we see that x could be 9 or -9, we cannot determine a unique value for y. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x - y = 2

We see that x = 2 + y, so, substituting, we have:

2 + y + y = 16

2y = 14

y = 7

Statement two alone is sufficient to answer the question.

Answer: B