Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 73

Chapter 1, Problem 73

In Exercises 69–82, simplify each complex rational expression. (1/x + 1/y)/(x+y)

Verified step by step guidance

Step 1: Identify the complex rational expression. The given expression is (1/x + 1/y) / (x + y). A complex rational expression is a fraction where the numerator or denominator (or both) contain fractions.

Step 2: Simplify the numerator. The numerator is 1/x + 1/y. To combine these fractions, find a common denominator, which is xy. Rewrite each fraction with the common denominator: (1/x = y/xy) and (1/y = x/xy). Combine them: (y + x)/xy.

Step 3: Rewrite the entire expression. Substitute the simplified numerator back into the original expression: ((y + x)/xy) / (x + y).

Step 4: Simplify the complex fraction. To divide fractions, multiply the numerator by the reciprocal of the denominator. The reciprocal of (x + y) is 1/(x + y). Multiply: ((y + x)/xy) * (1/(x + y)).

Step 5: Cancel common terms. Notice that (y + x) and (x + y) are equivalent, so they cancel out. The simplified expression is 1/xy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Rational Expressions

A complex rational expression is a fraction where the numerator, the denominator, or both contain rational expressions. Simplifying these expressions often involves finding a common denominator and reducing the fraction to its simplest form. Understanding how to manipulate these expressions is crucial for solving problems involving them.

Finding a Common Denominator

To simplify complex rational expressions, it is essential to find a common denominator for the fractions involved. This process allows for the combination of fractions into a single expression, making it easier to simplify. The common denominator is typically the least common multiple (LCM) of the denominators present in the expression.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms by dividing the numerator and denominator by their greatest common factor (GCF). This concept is fundamental in algebra, as it helps to present answers in a clearer and more concise form. Mastery of this skill is necessary for effectively handling complex rational expressions.

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