Problem 59 - College Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{1}{x}+\frac{1}{y}}{x+y} $$

Step-by-Step Solution

Answer

The simplified expression of the complex fraction \(\frac{\frac{1}{x}+\frac{1}{y}}{x+y}\) is \(\frac{1}{xy}\).

Step 1: Identify the Fractions

The complex fraction is made up by two smaller fractions in the numerator, which are \(\frac{1}{x}\) and \(\frac{1}{y}\), and one whole fraction in the denominator, which is \(x+y\). The goal is to simplify this expression.

Step 2: Simplify the Numerator

Apply the least common denominator to the fractions in the numerator. The least common denominator is the product of \(x\) and \(y\). Multiply each of the fractions by \(xy\) and put it over the common denominator. The new expression is \(\frac{y+x}{xy}\).

Step 3: Simplify the Whole Fraction

Now replace the numerator of the complex fraction with the expression found in step 2, which turns the complex fraction into \(\frac{\frac{y+x}{xy}}{x+y}\). Apply the distributive property here, the complex fraction turns to be \(\frac{1}{xy}\).

Key Concepts

Rational Expressions

Rational expressions are a lot like fractions, but they have polynomials in the numerator, the denominator, or both. These can be thought of as the "cousins" of regular fractions, as they involve variables and polynomial terms. For example, in the expression \( \frac{1}{x} \), \( x \) is a variable making it a rational number rather than a plain fraction. Understanding rational expressions is crucial because they often appear in algebra equations and calculus. When dealing with rational expressions, treat them similarly to simple fractions:

• Find common denominators when adding or subtracting.
• Factor where possible.
• Simplify as much as you can.

Remember, just like fractions, you cannot divide by zero. So make sure to check for restrictions on the variable that would make the denominator zero.

Simplification

Simplifying rational expressions is a must-have skill in algebra. This process involves making the expression as simple as possible, often by reducing the number of terms or determining equivalent simpler forms.The key in simplification is to find common factors or terms that can cancel out.For example, simplifying the expression \( \frac{\frac{1}{x} + \frac{1}{y}}{x+y} \) involves gathering the terms in the numerator over a common denominator and then simplifying further. In the expression from the Original Exercise:

1. Combine \( \frac{1}{x} \) and \( \frac{1}{y} \) to get \( \frac{x+y}{xy} \).
2. Divide this by the denominator \( x+y \), which becomes much simpler.

Simplification is about efficiency and clarity. It is making the expression look neater while preserving the expression's value.

Common Denominator

Finding a common denominator is one of the main steps when adding or subtracting fractions or rational expressions. In simple terms, a common denominator is just a number (or expression) that is the denominator for multiple fractions.The purpose is to rewrite each fraction so that they have the same bottom part, making them easier to combine.For instance, in the expression \( \frac{1}{x} + \frac{1}{y} \), the common denominator is \( xy \) since that is the smallest expression that both \( x \) and \( y \) can divide into without leaving a remainder.Using a common denominator allows us to write both fractions over \( xy \), making the sum straightforward. It allows the combination into a single fraction.

Distributive Property

The distributive property is a fundamental concept used frequently in algebra to simplify expressions. It states that \( a(b + c) = ab + ac \).This property allows us to distribute, or multiply, the term outside the parentheses by each term within the parentheses. In simplifying complex fractions, this property helps to break down expressions into those that are easier to manage. Taking the complex fraction \( \frac{\frac{y+x}{xy}}{x+y} \), we can see that the distributive property helps simplify the expression further by considering cancellations and evaluating the fractions. Think of the distributive property as unlocking the parenthesis so that each term inside gets multiplied separately.