Question:
Given that \( x \) and \( m \) are positive numbers, and \( m \) is a multiple of 3, compare the following quantities:
\[
\frac{x^m}{x^3} \quad \text{and} \quad \frac{m}{x^3}
\]
Given that \( x \) and \( m \) are positive numbers, and \( m \) is a multiple of 3, compare the following quantities:
\[
\frac{x^m}{x^3} \quad \text{and} \quad \frac{m}{x^3}
\]
Show Hint
When simplifying powers and exponents, make sure to carefully apply the laws of exponents to compare expressions.
Updated On: Oct 7, 2025
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given.
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The Correct Option is C
Solution and Explanation
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: \[ \boxed{\text{The correct answer is (3) The two quantities are equal.}} \]
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: \[ \boxed{\text{The correct answer is (3) The two quantities are equal.}} \]
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