If 2^(x + y) = 4^8, what is the value of y?

\(2^{(x + y)} = 4^8\)

Or, \(2^{(x + y)} = 2^{16}\)

So, \(x + y = 16\)

FROM STATEMENT - I ( INSUFFICIENT )

\(x^2 = 81\)

So, x = + 9

If x = +9 ; y = 7 & If x = - 9 ; y = 25

We do not have a unique value of y

FROM STATEMENT - II ( SUFFICIENT )

Given, \(x − y = 2\) and we know \(x + y = 16\) thus \(x = 9\) & \(y = 7\)

Thus, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked, answer will be (B)

To my understanding, a sqrt of a number can never be negative. How come the sqyt of 81 is 9 or -9?

Yes, when the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root. That is, \(\sqrt{81}=9\), NOT +9 or -9. Even roots have only a positive value on the GMAT.

In contrast, the equation \(x^2=81\) has TWO solutions, +9 and -9.

Given 2^(x+y)= 2^16;
so we have x+y=16 from q.
With x^2=81 we have 2 values for x +9 or -9. So y would also have 2 values.
So S1 is not sufficient.

From S2 given x-y=2; Solving for 2y=14; so y=2; Sufficient.

Ans:B

On the OG2018, 6.3 Practice Questions, Prob. 350. It appears as follows:

2^x +y = 4^8

But in the 6.5 Answer explanations, Prob 350. It appears as follows:

2^(x+y) = 4^8

The correct statement is the second one! I hope this helps someone!

Kudos +1 if it helps you!

2^(x+y) = 4^8 = (2^2)^8 = 2^16
=> x+y = 16

Q y = ?

S1:
x^2 = 81
x = +/- 9
If x = 9, y = 7
If x = -9 y = 25
NOT SUFFICIENT.

S2:
x-y=2
x = y+2
2y +2 = 16
y = 7
SUFFICIENT.

IMO B

If 2^(x + y) = 4^8, what is the value of y?

(1) x^2 = 81
(2) x − y = 2


The question asks if we can find Y or not. Immediately, before we start testing each statement, let's see if we can simplify this problem to make it easier on ourselves. 2^(x+y) = 4^8. Wouldn't it be easier if we could compare the two sides of the equation if they had the same base? Adjust accordingly. The given information turns into 2(x+y) = 2(^2^8) = 2^16. So we have this information: 2(x+y) = 2^16. But wait, we can go even further with the simplification. We know we are looking for Y. Each side has the same base (2), so we can simplify this into x+y=16. Now this is much more manageable.

Statement 1) x^2 = 81
We know that x can be either 9 or -9.
Let's plug each into our equation x+y=16, or y=16-x

X=9: y=16-(9)=7
X=-9: y=16-(-9)=25.

We get two different answers for y: 7 and 25, so this is an insufficient statement to determine what the value of y is.

Statement 2) x-y=2
We know that x+y=16 and x-y=2. We have two different 2-variable equations that aren't the same, so we can use whatever method we want (combination, substitution, case testing) to solve. I think combinations is the easiest, so let's solve:

x+y=16
+(x-y=2)
:2x=18, so we know that x=9. Plug this back into any of the two equations to find y. 9+y=16. y=7 This is sufficient, there is only one answer for y.

FYI - the 2020 GMAT official prep book has a typo in this problem, #406. The problem is shown as 2^(x)+y=4^8. I was so annoyed when I looked at the solution and the equation was supposed to be written 2^(x+y)=4^8.

Solution:

2x+y = (2^2)^8 = 2^16

=> x + y = 16

St (1)

x2 = 81 and thus

x = 9, - 9

Hence y = 7, 25

No unique value of y is obtained. Insufficient

St (2)

x-y =2

Along with x + y =16 (given) a unique value of y = 7 can be obtained.
Thus sufficient.

Hence option (b)

Hope this helps
Devmitra Sen (GMAT Expert-Quants)

2^(x+y) = 4^8
2^(x+y) = 2^2(8)
2^(x+y) = 2^16
x+y = 16

Statement 1
X^2+81
x= -9 or 9
x=-9 y=25
x=9 y=7

*Whenever you see a variable raised to an even exponent remember that that variable could be negative or positive.
Don't fall for the trap by assuming that x is positive.

Insufficient

Statement 2
x= 2+y
2+2y +16
y=7
Sufficient

Solution:

Question Stem Analysis:

We need to determine the value of y, given that 2^(x + y) = 4^8. Notice that 4^8 = 2^16. Therefore, x + y = 16. In other words, y = 16 - x.

Statement One Alone:

We see that x is either 9 or -9. If x = 9, then y = 16 - 9 = 7. However, if x = -9, then y = 16 - (-9) = 25. Since we can’t determine a unique value for y, statement one alone is not sufficient.

Statement Two Alone:

We see that x = y + 2. Since y = 16 - x, we have:

y = 16 - (y + 2)

y = 14 - y

2y = 14

y = 7

We see that y = 7. Statement two alone is sufficient.

Answer: B

Answer: Option B

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Great explantion BrentGMATPrepNow. One confusion why is it possible to use the derived info x + y = 16 with St 2 here but not possible in below question? Thanks Brent

https://gmatclub.com/forum/is-4-x-y-8-1 ... 39120.html

In the question above, we are GIVEN the information that 2^(x + y) = 4^8. In other words, this information is not provided in the statements.
From this given information, we are able to conclude that x + y = 16
In general, we can apply any given information to each of the statements.

In this question, https://gmatclub.com/forum/is-4-x-y-8-1 ... 39120.html, we are not GIVEN any information.
Here the question simply asks us "Is 4^(x+y) = 8^10?"

Hi Brent @BrentGMATPrepNow, thanks for your reply. still a bit confused...so basically....

GIVEN the information that 2^(x + y) = 4^8. , conclude x + y = 16, and use this to apply to the statement.
In this question, https://gmatclub.com/forum/is-4-x-y-8-1 ... 39120.html, given "Is 4^(x+y) = 8^10?"[/quote], conclude x+y=15 , but we can't use this to apply to statement? I'm not able to see any difference between the two questions here. Not sure what did I miss? Thanks Brent

, conclude x+y=15 , but we can't use this to apply to statement? I'm not able to see any difference between the two questions here. Not sure what did I miss? Thanks Brent [/quote]

The question above reads "If 2^(x + y) = 4^8, what is the value of y?"
Of the target question is: What is the value of y?
The underlined part, 2^(x + y) = 4^8, is given information that we can use to help determine whether each statement is sufficient to answer the target question.

In the question here, https://gmatclub.com/forum/is-4-x-y-8-1 ... 39120.html, we aren't given any information.
Instead we have the target question "Is 4^(x+y) = 8^10?"
So, without any extra information, there is no way to determine whether 4^(x+y) = 8^10
It COULD be the case that 4^(x+y) = 8^10, or it COULD be the case that 4^(x+y) ≠ 8^10
Since we don't know whether or not 4^(x+y) = 8^10, we can't then apply this information to each statement.