If a and b are integers, is a^5 < 4^b ? (1) a^3 = –27 (2) b^2 = 16

I thought statement 1 is sufficient. My answer is A.

Statement 2 is clearly insufficient.

Statement 1 establishes that a is negative.

No matter what integer you put as b in 4^b, it wouldnt return a negative value:
Ex:
4^4 = large positive number (256)
4^0 = 1 (still positive)
4^-2 = small fraction, but still positive (1/16)

Thus proving the inequality sound.

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Target question: Is a^5 > 4^b

Statement 1: a³ = -27
Solve to get: a = -3
So, a^5 = (-3)^5 = -243
Since 4^b will be POSITIVE for all values of b, the answer to the target question is NO, a^5 is definitely NOT greater than 4^b
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: b^2 = 16
Solve to get: EITHER b = 4 OR b = -4
Let's test some possible cases:
Case a: a = 1 and b = 4. In this case, a^5 = 1^5 = 1, and 4^b = 4^4 = 256. Here, the answer to the target question is NO, a^5 is definitely NOT greater than 4^b
Case b: a = 10 and b = 4. In this case, a^5 = 10^5 = 100,000, and 4^b = 4^4 = 256. Here, the answer to the target question is YES a^5 is greater than 4^b
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

From S1:

\(a^{-3} = 27\)
a = -3
But, No info about b.
Insufficient.

From S2:

\(b^2 = 16\)
b = +4 or -4
No info about a.
Insufficient.

Combining both:

-243 < 4^4
-243 < 256
Yes \(a^5 < 4^b\)
If b = -4
\(-243 < 4^{-4}\)
-243 < 1/256
Yes \(a^5 < 4^b\)

C is the answer.

Can you give me a scenario where 4^b will yield a negative value... I feel that A is sufficient because no matter what value b takes, 4^b will always be greater than 0 ...

Take extreme case , b = -1000/3 .. 4^-1000/3 = 4^-1/3(1000) = (1/4^1/3)^1000... I need to know what's a and statement 1 gives me value of a... Or atleast sign of a. Which is sufficient to tell that LHS<RHS... I would go with a... Am I missing something??

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Can you provide me a scenario where 4^b will yield a negative value... I have been trying soo hard to figure out the RHS...

My reasoning -

A look at RHS - 4^b... If b = 0.. 4^0 = 1, if b = 1/2.. 4^1/2 = 2.. if b = -1/2 ... 4^-1/2 = 1/4^1/2 = 1/2... Take extreme case b= -1000/3 .. then 4^-1000/3 = 4^-1/3(1000)= {1/4^1/3}^1000....in every case, no Matter whether b is a negative integer, a positive integer, a fractional value... I get 4^b > 0... Soo I just need to know whether a is positive or negative... By statement 1 , a is negative... Which means LHS is <0 ... Which implies that LHS<RHS... sufficient...

In statement 2 - b^2 = 16... As explained earlier 4^b > 0... Nothing is mentioned about a... If a is negative integer, LHS < RHS... If a is positive integer LHS>RHS... I get two possibilities... Not sufficient... I would go with A... I don't know whether I missed out something.... I would appreciate if someone explain me what I did wrong??...

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Yes, correct. I missed it

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If base is negative & power is odd, resulting number must be negative.

And if base is positive, whatever the power, the resulting number must be POSITIVE.
Statement one is SUFFICIENT.

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Learning points:
If BASE is POSITIVE whatever the power (positive/negative/zero), the resulting number must be POSITIVE.

If base is Negative, there are two situations:
1. If power is ODD, the resulting number is NEGATIVE.
2. if power is EVEN, the resulting number is POSITIVE.


That means, there is a SINGLE situation in which the resulting number is NEGATIVE,
BASE IS NEGATIVE & POWER IS ODD.
In all other cases, the resulting number is POSITIVE.

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NEGATIVE POWER implies that the resulting number is FRACTION. (NEGATIVE POWER DOES NOT SAY ANYTHING WHETHER the resulting number is NEGATIVE or POSITIVE.)

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We need to determine whether a^5 is less than 4^b.

Statement One Alone:

a^3 = -27

We see that a is -3; so a^5 is negative. However, 4^b is always positive regardless of the value of b. Therefore, a^5 is indeed less than 4^b. Statement one alone is sufficient to answer the question.

Statement Two Alone:

b^2 = 16

So we see that b = 4 or -4.

However, without knowing anything about the value of a, we cannot determine whether a^5 is less than 4^b. For example, if a < 0, then a^5 < b^4. However, if a > 0, then a^5 might be greater than b^4 (for example, a = 5). Statement two alone is not sufficient to answer the question.


Answer: A
_________________

Hi All,

We're told that A and B are INTEGERS. We're asked if A^5 is less than 4^B. This question can be approached with a mix of Number Properties and TESTing VALUES. To start, it's worth noting that raising +4 to ANY power will lead to a POSITIVE value (for example, 4^0 = 1, 4^-1 = 1/4, etc.).

(1) A^3 = -27

Fact 1 tells us that A = -3, although once you realize that A is a NEGATIVE value, you can stop working. By extension, A^5 would be a NEGATIVE value - and since 4^B is a POSITIVE value, we know that A^5 will ALWAYS be less than 4^B. Thus the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

(2) B^2 = 16

Fact 2 tells us that B = 4, but we don't know anything about the value of A. 4^4 = 16^2 = 256
IF...
A = 1, then A^5 = 1 and the answer to the question is YES.
A = 4, then 4^5 is clearly GREATER than 4^4, so the answer to the question is NO.
Fact 2 is INSUFFICIENT

Final Answer:

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

Hi Chethan92,

This is incorrect. Statement 1 is clearly sufficient.

As a = -3, -3^5 will be negative (as raised to odd power) and 4^b is positive irrespective of the value of b.

(1) Gives a definate negative value of a and exponent is odd so it will give a negative value. Any value for b will give a positive number. A is Sufficient.
(2) Give two values +, - 4 for b but no informamtion about a. Insufficient.

Ans. A.
_________________

(1) a is negative and its exponent is odd but the base of other side is is positive so even the power negative the result will be positive and the answers will be (definite) Yes. Sufficient.

(2) We know don't know about a. Insufficient.

The answer is A.
_________________

Video solution from Quant Reasoning:
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Answer: Option A

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Thanks BrentGMATPrepNow. To clarify regaarding statement 2, for b = +/-4 is already insufficient as it will yield different results? Thanks for your time in advanced.

Great question.
Even though we have two possible values for b, we must still has those values with regard to the target question.
For example, if the target question were "Is b greater than -10?" the two possible values for b (4 or -4) would yield the same answer to the target question (YES, b is greater than -10), in which case statement 2 would be sufficient.

Here is a related question: https://gmatclub.com/forum/what-is-the- ... 35607.html

I hope that helps

Brilliant thanks BrentGMATPrepNow for the clarification and make sense especially with the example link.
In your Statement 2 of Case a & b testing, any reason of testing only with 4 in both cases and not 4 and -4 each? Thanks Brent.