Problem 75 - Algebra A Combined Function

Question

Multiply or divide as indicated. $$ \left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}} \div \frac{x^{2}-y^{2}}{3 x}\right) \cdot \frac{x^{2}+y^{2}}{6} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{x}{2}\).

Step 1: Understand the Complex Fraction

You are given a complex fraction expression, \[ \left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}} \div \frac{x^{2}-y^{2}}{3 x}\right) \cdot \frac{x^{2}+y^{2}}{6} \]. This means you need to perform a division followed by a multiplication.Start by simplifying inside the parentheses.

Step 2: Simplify the Division

Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, rewrite the division as multiplication:\[ \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \cdot \frac{3x}{x^{2}-y^{2}} \]Now cancel \(x^{2}-y^{2}\) from numerator and denominator:\[ \frac{3x}{x^{2}+y^{2}} \].

Step 3: Perform the Multiplication

Now multiply the simplified expression by the remaining fraction:\[ \frac{3x}{x^{2}+y^{2}} \cdot \frac{x^{2}+y^{2}}{6} \]The \(x^{2}+y^{2}\) terms cancel out, resulting in:\[ \frac{3x}{6} \].

Step 4: Simplify the Result

Finally, simplify the fraction \(\frac{3x}{6}\) by dividing both numerator and denominator by 3:\[\frac{3x}{6} = \frac{x}{2}\].

Key Concepts

Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both are also fractions.
This can make them appear challenging at first since they involve multiple layers of division.
In algebra, complex fractions often appear in problems with rational expressions.

To simplify a complex fraction, it's important to first understand the structure.
Consider each fraction within the numerator and denominator separately.
Start simplifying step by step, often by applying operations such as multiplication or division.
You may also need to rearrange the expression to make calculations easier.

Using the principle of keeping it simple, always try to imagine the complex fraction as a simpler, single fraction.
This helps in reducing errors and eases mental computation.

Fraction Simplification

Simplifying fractions is crucial in algebra to make expressions more manageable.
The goal is to write the fraction in its simplest form, where the numerator and denominator have no common factors other than 1.

Steps to simplify fractions include:

• Identifying common factors in the numerator and the denominator.
• Dividing both by their greatest common factor.

When simplifying the fraction, ensure that you don't overlook common terms in polynomials.
Sometimes, polynomials in rational expressions can be factored out to reveal common factors.

In problems involving expressions like polynomials, special identities like difference of squares or binomial squares are invaluable for factorization.
This makes it easier to identify and cancel out common factors.

Polynomial Division

Polynomial division is a method used to simplify expressions where polynomials are divided by other polynomials.
Just like with numbers, this process aims to find how many times one polynomial can be contained in another.

The division of polynomials can be carried out using long division, akin to the long division of numbers.
Alternatively, synthetic division can be employed, which is usually quicker.
In algebraic simplification, division by polynomials often involves factoring both the numerator and the denominator first.

For example, recognize if a polynomial is a difference of squares, like in the exercise, \(x^2-y^2 = (x+y)(x-y)\).
Understanding these patterns allows you to break down complex expressions and simplify polynomial division significantly.
Practicing polynomial division not only simplifies algebraic expressions but also strengthens understanding of polynomial behavior.

Reciprocal in Fractions

The reciprocal of a fraction is simply the fraction flipped, meaning the numerator and the denominator are switched.
Finding the reciprocal is crucial in division of fractions.
When dividing by a fraction, multiply by its reciprocal instead.
For instance, division like \( \frac{a}{b} \div \frac{c}{d} \) becomes multiplication \( \frac{a}{b} \cdot \frac{d}{c} \).

This principle stems from understanding that dividing by a fraction is akin to finding how many times the fraction fits into another number.
The reciprocal thus makes this process simple.
In algebra, flipping the fraction to its reciprocal is usually one of the first steps when facing division operations.
It's a straightforward yet powerful tool that turns the complicated division operation into a much simpler multiplication.
Always check your work to ensure nothing but the intended terms have been reciprocated.