Question

If \( mx^n - nx^m = 0 \), then what is the value of \( \frac{1}{x^m + x^n} + \frac{1}{x^m - x^n} \) in terms of \( x \)?

• A. \( \frac{2mn}{x^n(m^2 - n^2)} \)
• B. \( \frac{2mn}{x^n(n^2 - m^2)} \)
• C. \( \frac{2mn}{x^m(m^2 - n^2)} \)
• D. \( \frac{2mn}{x^m(n^2 - m^2)} \)

Hint:

Use the given equation to substitute \( x^m \) or \( x^n \) and simplify.

Answer 1

The Correct Option is A

From \( mx^n = nx^m \Rightarrow \frac{m}{n} = x^{m-n} \Rightarrow x^{m-n} = \frac{m}{n} \). Use algebraic manipulation to express the required sum in terms of \( x^n \), then rationalize to find the result. \[ { \frac{2mn}{x^n(m^2 - n^2)} } \]