Problem 74 - Algebra and Trigonometry

Question

Simplify the compound fractional expression. $$ \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(x+y)^2}{xy} \).

Step 1: Simplify the Numerator

The numerator of the compound fraction is \( x^{-1} + y^{-1} \). Rewriting each term using positive exponents, we get: \( \frac{1}{x} + \frac{1}{y} \). To combine these fractions, find a common denominator, which is \( xy \). Thus, the expression becomes: \( \frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy} \).

Step 2: Simplify the Denominator

The denominator of the compound fraction is \( (x+y)^{-1} \). Rewriting using a positive exponent, it becomes \( \frac{1}{x+y} \).

Step 3: Simplify the Entire Fraction

The original compound fraction is \( \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} \). Substitute the simplified expressions for the numerator and denominator from Steps 1 and 2: \( \frac{\frac{y+x}{xy}}{\frac{1}{x+y}} \). This is equivalent to multiplying by the reciprocal: \( \frac{y+x}{xy} \times \frac{x+y}{1} \).

Step 4: Simplify the Expression

The expression from Step 3 simplifies to: \( \frac{(y+x)(x+y)}{xy} \). Recognize that \( y+x = x+y \), which results in \( \frac{(x+y)^2}{xy} \). Since \( y+x \) and \( x+y \) represent the same values, this step confirms the expression is \( \frac{(x+y)^2}{xy} \).

Key Concepts

Compound Fractions

Compound fractions, often referred to as complex fractions, are fractions that have fractions in either the numerator, the denominator, or both. They can look intimidating at first, but with a few key steps, you can simplify them easily. To begin simplifying a compound fraction, tackle the numerator and denominator separately. In the case of \( \frac{x^{-1} + y^{-1}}{(x+y)^{-1}} \), you'll work on both parts one after the other.

• Simplify the numerator to a single fraction by combining fractions using a common denominator.
• Simplify the denominator as a single fraction, if applicable.
• Combine the simplified numerator and denominator, turning the division of fractions into multiplication using reciprocals.

These steps will lead you to a simpler expression, making it easier to work with or solve.

Positive Exponents

Positive exponents make life easier when simplifying expressions, especially with variables. In mathematics, exponents indicate repeated multiplication. A positive exponent shows how many times to multiply the base number by itself. When you encounter negative exponents, like \( x^{-1} \), change them to positive exponents to simplify the process. For instance, \( x^{-1} = \frac{1}{x} \), which turns the expression into a more manageable form.

• Convert negative exponents into positive by flipping to the reciprocal (\( x^{-n} = \frac{1}{x^n} \)).
• Simplifying with positive exponents often means fewer steps and clearer results, avoiding potential for errors.

Rewriting without negatives can illuminate the path to the solution, especially in complicated algebraic expressions.

Common Denominator

Finding a common denominator is a crucial step when adding or subtracting fractions. This process ensures that the fractions involved have the same bottom number, allowing them to be combined more readily. In our example, the expression \( x^{-1} + y^{-1} \) became \( \frac{1}{x} + \frac{1}{y} \). To add these fractions, you need a common denominator, which would be the least common multiple of \( x \) and \( y \), or \( xy \).

• Multiply each fraction by the missing part of the common denominator to equalize them.
• So \( \frac{1}{x} = \frac{y}{xy} \) and \( \frac{1}{y} = \frac{x}{xy} \).
• Combine the fractions: \( \frac{y+x}{xy} \).

After aligning the fractions, combining them is straightforward, leaving you with a single simplified fraction to work with.

Reciprocals

Reciprocals are essential in simplifying compound fractions, particularly when you have to deal with division in the fractions.The reciprocal of a number or a fraction is what you multiply by to get a product of one. In terms of fractions, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).In our problem, after achieving simple fractions, you multiply by the reciprocal to transform division into multiplication.

• For the compound fraction \( \frac{\frac{y+x}{xy}}{\frac{1}{x+y}} \), multiply by the reciprocal of the denominator.
• This means \( \frac{y+x}{xy} \times (x+y) \), getting rid of the complex division.

Using reciprocals streamlines the operation, converting intricate expressions into clear, workable solutions.