Question

If m and n are positive integers, is \((\sqrt{m})^n\) an integer?
(1) \((\sqrt{m})\) is an integer.
(2) \((\sqrt{n})\) is an integer.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

When dealing with exponents and roots, rewrite the expression in fractional exponent form (e.g., \(\sqrt{m} = m^{1/2}\)). This often makes the rules of exponents easier to apply and the conditions for integer results clearer.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks whether the expression \((\sqrt{m})^n\) results in an integer, given that \(m\) and \(n\) are positive integers. The expression can also be written as \(m^{n/2}\). For this to be an integer, we generally need either \(m\) to be a perfect square or \(n\) to be an even number.
Step 2: Detailed Explanation:
Analyze Statement (1): \((\sqrt{m})\) is an integer.
Let's say \(\sqrt{m} = k\), where \(k\) is an integer.
The expression in the question becomes \(k^n\).
Since \(k\) is an integer and \(n\) is a positive integer, the result of raising an integer to a positive integer power (\(k^n\)) will always be an integer.
For example, if \(m=9\) (\(\sqrt{m}=3\)) and \(n=5\), then \((\sqrt{9})^5 = 3^5 = 243\), which is an integer.
The answer to the question is always "Yes". Therefore, statement (1) is sufficient.
Analyze Statement (2): \((\sqrt{n})\) is an integer.
This means that \(n\) is a perfect square. So, \(n\) can be 1, 4, 9, 16, etc. The question is whether \((\sqrt{m})^n\) is an integer.

Case 1 (Answer "Yes"): Let \(n=4\) (a perfect square) and \(m=5\). The expression is \((\sqrt{5})^4 = (5^{1/2})^4 = 5^2 = 25\). This is an integer. The answer is "Yes". (Note: This works for any even value of n, and perfect squares can be even).
Case 2 (Answer "No"): Let \(n=9\) (a perfect square) and \(m=2\). The expression is \((\sqrt{2})^9 = 2^{9/2} = 2^{4.5}\). This is not an integer. The answer is "No". (Note: This shows that if n is an odd perfect square, the result depends on m).
Since we can get both "Yes" and "No" answers, statement (2) is not sufficient.
Step 3: Final Answer:
Statement (1) alone is sufficient, but statement (2) alone is not.