Question

Given that \( x \) and \( m \) are positive numbers, and \( m \) is a multiple of 3, compare the following quantities: \[ \frac{x^m}{x^3} \quad \text{and} \quad \frac{m}{x^3} \]

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given.
• E.

Hint:

When simplifying powers and exponents, make sure to carefully apply the laws of exponents to compare expressions.

Answer 1

The Correct Option is C

Step 1: Simplify the expressions.
We are asked to compare \( \frac{x^m}{x^3} \) and \( \frac{m}{x^3} \). Since \( m \) is a multiple of 3, let \( m = 3k \) for some integer \( k \). Thus, \( \frac{x^m}{x^3} = x^{m-3} = x^{3k-3} = x^{3(k-1)} \).
Step 2: Compare the quantities.
Now, we are comparing \( x^{3(k-1)} \) with \( \frac{m}{x^3} = \frac{3k}{x^3} \). Since the relationship involves powers of \( x \) and a coefficient \( m \), we find that for any values of \( k \), the quantities are equal.
Step 3: Conclusion.
Thus, the two quantities are equal.
Final Answer: \[ \boxed{\text{The correct answer is (3) The two quantities are equal.}} \]