Problem 37 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{s}{r}+\frac{r}{s}}{\frac{s}{r}-\frac{r}{s}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{s^2 + r^2}{s^2 - r^2} \).

Step 1: Identifying the Numerator and Denominator

The complex fraction is \( \frac{ \frac{s}{r} + \frac{r}{s} }{ \frac{s}{r} - \frac{r}{s} } \). The numerator is \( \frac{s}{r} + \frac{r}{s} \) and the denominator is \( \frac{s}{r} - \frac{r}{s} \).

Step 2: Find a Common Denominator for the Numerator

For \( \frac{s}{r} + \frac{r}{s} \), find a common denominator, which is \( rs \). Rewrite \( \frac{s}{r} \) as \( \frac{s^2}{rs} \) and \( \frac{r}{s} \) as \( \frac{r^2}{rs} \). Now the expression becomes \( \frac{s^2 + r^2}{rs} \).

Step 3: Find a Common Denominator for the Denominator

For \( \frac{s}{r} - \frac{r}{s} \), use the common denominator \( rs \) as well. Rewrite \( \frac{s}{r} \) as \( \frac{s^2}{rs} \) and \( \frac{r}{s} \) as \( \frac{r^2}{rs} \). Now the expression becomes \( \frac{s^2 - r^2}{rs} \).

Step 4: Simplifying the Complex Fraction

The complex fraction is now \( \frac{ \frac{s^2 + r^2}{rs} }{ \frac{s^2 - r^2}{rs} } \). Since both the numerator and the denominator share the same common denominator \( rs \), they cancel out. Thus, the expression becomes \( \frac{s^2 + r^2}{s^2 - r^2} \).

Step 5: Final Simplification

The expression \( \frac{s^2 + r^2}{s^2 - r^2} \) cannot be simplified using basic algebraic identities (e.g., difference of squares), so this is the simplified form of the original complex fraction.

Key Concepts

Numerator and Denominator

In a complex fraction, it's crucial to distinguish between the numerator and the denominator. These are the two main parts that form any fraction, whether simple or complex.

• **Numerator:** This is the top part of the fraction. It represents the number of parts we have.
• **Denominator:** This is the bottom part, indicating the number of equal parts the whole is divided into.

For our given problem, the complex fraction is \( \frac{ \frac{s}{r} + \frac{r}{s} }{ \frac{s}{r} - \frac{r}{s} } \). Here, the numerator is \( \frac{s}{r} + \frac{r}{s} \) and the denominator is \( \frac{s}{r} - \frac{r}{s} \). Understanding these parts is the first step towards simplifying the expression effectively.

Common Denominator

Finding a common denominator is a fundamental step in adding or subtracting fractions. It simplifies complex expressions by providing a shared base for comparison.

• The **common denominator** is generally the least common multiple of the denominators, allowing you to combine fractions more easily.
• In the exercise, the fractions \( \frac{s}{r} \) and \( \frac{r}{s} \) are involved, where \( rs \) serves as the common denominator.

Expressing \( \frac{s}{r} \) as \( \frac{s^2}{rs} \) and \( \frac{r}{s} \) as \( \frac{r^2}{rs} \) gives you a common base to work with, thus allowing you to rewrite the complex fraction in simpler terms.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This includes combining terms, canceling out common factors, and leveraging algebraic identities.

• For the given problem, rewriting both the numerator \( \frac{s}{r} + \frac{r}{s} \) and the denominator \( \frac{s}{r} - \frac{r}{s} \) with the common denominator \( rs \) gives us \( \frac{s^2 + r^2}{rs} \) and \( \frac{s^2 - r^2}{rs} \) respectively.
• By simplifying \( \frac{ \frac{s^2 + r^2}{rs} }{ \frac{s^2 - r^2}{rs} } \), we use the property of fractions where divided numerators and denominators can have their common denominators canceled, resulting in \( \frac{s^2 + r^2}{s^2 - r^2} \).

This leaves us with a simpler fraction devoid of the complexity of the original nested structure.

Difference of Squares

The difference of squares is a specific algebraic identity useful for simplifying certain expressions. It states that \( a^2 - b^2 = (a - b)(a + b) \).

• This identity is relevant for the denominator of the simplified fraction \( \frac{s^2 + r^2}{s^2 - r^2} \), which is \( s^2 - r^2 \).
• While the numerator does not fit this identity (as it contains a sum, not a difference), acknowledging the form \( s^2 - r^2 \) helps in recognizing potential factorization strategies if needed further.

Although in this case, the expression \( \frac{s^2 + r^2}{s^2 - r^2} \) cannot be further simplified using the difference of squares alone, recognizing such identities can be invaluable in tackling similar problems in future mathematical exercises.