Problem 60 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} $$

Step-by-Step Solution

Answer

\( \frac{(x+y)^2}{xy} \)

Step 1: Interpret the Problem

The complex fraction we need to simplify is \( \frac{x^{-1} + y^{-1}}{(x+y)^{-1}} \). Each part involves negative exponents, which we need to handle carefully.

Step 2: Simplify the Numerator

The numerator is \( x^{-1} + y^{-1} \). Using the property \( a^{-1} = \frac{1}{a} \), rewrite the numerator as \( \frac{1}{x} + \frac{1}{y} \). To add these fractions, find a common denominator: \( \frac{y}{xy} + \frac{x}{xy} = \frac{y+x}{xy} \).

Step 3: Simplify the Denominator

The denominator is \( (x+y)^{-1} \), which is equivalent to \( \frac{1}{x+y} \) by using the negative exponent rule \( a^{-1} = \frac{1}{a} \).

Step 4: Divide by a Fraction

To divide \( \frac{y+x}{xy} \) by \( \frac{1}{x+y} \), multiply by the reciprocal: \( \frac{y+x}{xy} \times \frac{x+y}{1} = \frac{(y+x)(x+y)}{xy} \). Simplify \( (y+x)(x+y) = (x+y)^2 \).

Step 5: Final Result

The simplified form of the complex fraction is \( \frac{(x+y)^2}{xy} \). There's no further simplification possible.

Key Concepts

Negative Exponents

Negative exponents can initially seem puzzling, but they truly simplify into a concept of reciprocal expressions. When we encounter a term with a negative exponent, like \(x^{-1}\), it can be rewritten as \(\frac{1}{x}\). This transformation holds because the negative exponent implies the reciprocal.

The rule \(a^{-n} = \frac{1}{a^n}\) tells us that the base \(a\) moves to the denominator, converting the exponent from negative to positive. Knowing this rule helps simplify many math problems, including this particular complex fraction.

Common Denominator

When adding fractions, a crucial step is identifying a common denominator. This makes the addition of fractions seamless by aligning the denominators.

For instance, in the expression \(\frac{1}{x} + \frac{1}{y}\), the denominators are \(x\) and \(y\). To add these fractions effectively:

• Find the least common multiple of the denominators, which is \(xy\).
• Re-express each fraction with the common denominator. Here: \(\frac{y}{xy} + \frac{x}{xy} = \frac{y+x}{xy}\).

By employing a common denominator, you streamline the process of adding fractions, which is vital in solving complex fractions.

Reciprocal Multiplication

Dividing by a fraction is similar to multiplying by its reciprocal. When you see mathematics involving division of fractions, flip the second fraction and switch the operation to multiplication.

In our problem, after simplifying both the numerator and denominator, you have \(\frac{y+x}{xy}\) divided by \(\frac{1}{x+y}\). By flipping the denominator and changing division into multiplication, it becomes:

• \(\frac{y+x}{xy} \times \frac{x+y}{1} = \frac{(y+x)(x+y)}{xy}\)

This results in a much more manageable multiplication expression, helping to simplify the complex fraction further.

Fraction Addition

Adding fractions can be straightforward once you’re comfortable with finding a common denominator. It’s like finding a shared language for fractions to "speak" with each other.

In our exercise, once you've rewritten \(x^{-1}\) and \(y^{-1}\) as \(\frac{1}{x}\) and \(\frac{1}{y}\), respectively:

• Express each with the common denominator \(xy\).
• This allows you to add them directly as \(\frac{y}{xy} + \frac{x}{xy}\).

The sum becomes \(\frac{y+x}{xy}\), making the problem easier to tackle. Mastering fraction addition lays the groundwork for understanding more intricate topics like complex fractions.