What is the value of k?

1.\(2^k\) = \(\frac{1}{8}\)

Or, \(2^k\) = \(2^-^3\)

Hence k = -3

2. \(4^k+3 = 1\)

Or,\(2^2^k\)\(+3 = 1\)

Or,\(2^2^k\)\(= -2\)

Or,\(2^2^k\)\(= -1(2)\)

Or, 2k = 1

So, K = 1/2

So, (D) EACH statement ALONE is sufficient.

IMHO (D)
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Please format the question properly and provide the OA when posting.



What is the value of k?

(1) 2^k = 1/8 --> 2^k = 2^(-3) --> k = -3. Sufficient.

(2) 4^(k + 3) = 1 --> k + 3 = 0 --> k = -3. Sufficient.

Answer: D.
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Thank you guys! This was very quick.

Hi Nadiuska,

Since the other posters have already correctly answered the question, I won't rehash any of that work here. Instead, I'll elaborate a bit more on the Exponent rules involved. As a category, you won't see too many Exponent rule prompts on Test Day (probably 2-3, not counting squared-terms or Quadratics).

In this DS prompt, the two Facts test your knowledge of 2 specific exponent rules:

1) What happens when you raise a base to a NEGATIVE exponent.
2) What happens when you raise a base to a 0 exponent.

For the first rule, when raising a base to a NEGATIVE power, you essentially "flip" the calculation and turn the negative into a positive...

\(3^{-2}\) = 1/\(3^{2}\)= 1/9

For the second rule, when an exponent-based calculation = 1, you have a few different possibilities:
1) The base = 1, so the exponent could be anything.
2) The base = -1 and the exponent is an even integer.
3) The exponent is 0, so the base could be anything.

While knowing these rules won't lead to a lot of points on Test Day, they can help you to pick up a couple of additional correct answers. As you string together enough of the rarer question types, you can see a nice bump-up in your score.

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

Thank you very much Empowergmat for the extra explanation.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of k?

(1) 2^k = 1/8
(2) 4^(k + 3) = 1

There is one variable (k) and 2 equations are given by the conditions, so there is high chance (D) will be our answer.
For condition 1, 2^k=1/8=2^(-3), k=-3. This is sufficient.
For condition 2, 4(k+3)=1 =4^0, k+3=0, k=-3. This is also sufficient.
The answer become (D).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

1. 2^k = 2^-3

2. k+3=0, k=-3
Each independently sufficient: D

Hi Bunuel,

Would you please explain the exponent rules that you applied to statement 2? I don't understand how you got k+3 = 0 from 4^(k+3) = 1.

My understanding is that when there is addition in exponents on the same base number, then it can be written as 4^(k+3) --> 4^k * 4^3. Then divide both sides by 4^3 to get 1/64 on the right side of the equation and then 4^k = 4^-3, so k = -3.

What am I missing/is there a short cut rule you are applying here?

Thanks for your help!

Hello jmwon

Rules-

a^x = a^y Then x=y provided a!=1...rule#1

anything^0 = 1...rule#2

No let us see statement#2
4^(K+3) = 1
Can we write 1 as 4^0 (rule#2)? Yes, so


4^(K+3) = 4^0...from rule#1
Thus,

k+3=0
k=-3. Hope this helps.

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