Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find a common denominator for the fractions in the numerator

Here, both fractions in the numerator have different denominators, \(x\) and \(y\). So, the LCD (Least Common Denominator) will be \(xy\). The LCD is then multiplied with each fraction. That simplifies \(\frac{1}{x}+\frac{1}{y}\) to \(\frac{y}{xy} + \frac{x}{xy}\) = \(\frac{x+y}{xy}\)

Step 2: Recreate the complex fraction

In this step, replace the original numerator \(\frac{1}{x}+\frac{1}{y}\) with \(\frac{x+y}{xy}\) which was done in step one. Now, the complex function becomes \(\frac{\frac{x+y}{xy}}{x+y}\).

Step 3: Simplify the complex fraction

To simplify, multiply the whole expression by the reciprocal of the denominator. This cancels out the \(x+y\) terms on top and bottom of the fraction. Therefore, the fraction simplifies to \(\frac{1}{xy}\)

Key Concepts

Understanding Algebra: The Basics

Algebra is the language of mathematics that helps us express patterns and relationships using numbers and symbols. Think of it as a way of solving problems by finding the unknowns. In this exercise, we use algebraic techniques to simplify a complex rational expression. Complex rational expressions often include fractions within fractions. These can look tricky, but with a systematic approach, they become manageable.

When working with algebra, it's essential to be comfortable with:

• Variables, such as \(x\) and \(y\), which can hold any number.
• Operations, including addition, subtraction, multiplication, and division.
• Manipulating expressions to find a solution.

The goal is to simplify expressions into their most reduced form. This makes calculations easier and clearer.

Finding the Least Common Denominator

The Least Common Denominator (LCD) is critical when dealing with fractions. It helps you combine fractions into a single expression. When you have fractions with different denominators, like \(\frac{1}{x}\) and \(\frac{1}{y}\), finding the LCD involves identifying the smallest common multiple of these denominators.

For our exercise, the denominators are \(x\) and \(y\). The smallest multiple that both can divide is \(xy\). By converting fractions to a common denominator, you make it easy to add or subtract them. Here's a quick process:

• Multiply each fraction's numerator and denominator by a term that would create the LCD.
• For \(\frac{1}{x}\), multiply by \(\frac{y}{y}\) to get \(\frac{y}{xy}\).
• For \(\frac{1}{y}\), multiply by \(\frac{x}{x}\) to get \(\frac{x}{xy}\).
• Add these fractions together to form \(\frac{x+y}{xy}\).

With the LCD, you've rewritten the expression in a way that makes it straightforward to simplify.

Simplifying Fractions: The Key Steps

Simplifying fractions involves reducing them to their simplest form so that they are more understandable and easier to work with. In our complex rational expression, this means transforming \(\frac{\frac{x+y}{xy}}{x+y}\) into something simpler. Here’s how you can think about it:

• You have a fraction within a fraction – the numerator is already simplified using the LCD.
• To simplify the entire expression, multiply by the reciprocal of the denominator. The denominator is \(x+y\).
• Multiplying by the reciprocal \(\frac{1}{x+y}\) cancels out the \(x+y\) terms.
• The result is \(\frac{1}{xy}\), a simpler and clearer expression.

When simplifying, look to cancel out terms by multiplication or division to reach the most reduced form. This makes evaluating and interpreting expressions much easier. Remember, practice makes perfect with these concepts, so keep trying different problems to grow your confidence!