Question

Is \(a^{44} < b^{11}\), given that a = 2 and b is an integer?
A. b is even
B. b is greater than 16

• A. The questions can be answered by one of the statements, but not by the other
• B. The question can be answered using either of the two statements alone.
• C. The questions can be answered using both the statements together, but cannot be answered using either statement alone.
• D. The question cannot be answered even by using both the statements together.

Hint:

Write \(2^{44}\) as something raised to the power 11, then compare the bases. Remember that 11 is an odd power.

Answer 1

The Correct Option is A

Step 1: Rewrite the left side so that both sides have the same power.
Since a = 2, the left side is \(a^{44} = 2^{44}\).
The power 44 equals \(4 \times 11\), so \(2^{44} = (2^4)^{11} = 16^{11}\).
So the question becomes: is \(16^{11} < b^{11}\)?

Step 2: Reduce the comparison to the bases.
The power 11 is odd. For an odd power, a bigger number always has a bigger power, and this holds for negative numbers too. For example \((-3)^{11} < (-2)^{11}\).
So \(16^{11} < b^{11}\) is true exactly when \(16 < b\).
The whole question is therefore: is the integer b greater than 16?

Step 3: Check statement A (b is even).
Take b = 2. It is even, but 2 is not greater than 16, so the answer to the question is NO.
Take b = 18. It is even and greater than 16, so the answer is YES.
Statement A allows both answers, so A alone is NOT sufficient.

Step 4: Check statement B (b is greater than 16).
This is exactly the condition we reduced the question to. If b is greater than 16, then \(b^{11} > 16^{11} = a^{44}\), so the answer is always YES.
So B alone is sufficient.

Step 5: Match with the four choices.
Choice 1 says the question can be answered by one statement but not by the other. This fits: B works, A does not.
Choice 2 is wrong because A cannot answer the question alone.
Choice 3 is wrong because B alone is already enough, so we do not need both together.
Choice 4 is wrong because the question can be answered.

Final Answer:
Only statement B decides the question, so the correct choice is option 1. \[ \boxed{1} \]