Questions: Simplify the complex fraction. [ (n/d + d/n) / (n/d - d/n) ]

Solution Steps

To simplify the given complex fraction, we need to follow these steps:

1. Combine the fractions in the numerator and the denominator separately.
2. Simplify the resulting fractions.
3. Divide the simplified numerator by the simplified denominator.

We start with the complex fraction:

\[ \frac{\frac{n}{d} + \frac{d}{n}}{\frac{n}{d} - \frac{d}{n}} \]

The numerator can be combined as follows:

\[ \frac{n}{d} + \frac{d}{n} = \frac{n^2 + d^2}{nd} \]

The denominator can be combined as:

\[ \frac{n}{d} - \frac{d}{n} = \frac{n^2 - d^2}{nd} \]

Substituting the combined forms back into the complex fraction gives us:

\[ \frac{\frac{n^2 + d^2}{nd}}{\frac{n^2 - d^2}{nd}} = \frac{n^2 + d^2}{n^2 - d^2} \]

Final Answer