Problem 67 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{t}{x^{2}-y^{2}}}{\frac{t}{x+y}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{1}{x-y} \).

Step 1: Identify the Structure

The given expression is a complex fraction of the form \( \frac{\frac{a}{b}}{\frac{c}{d}} \), where \( a = t \), \( b = x^{2} - y^{2} \), \( c = t \), and \( d = x+y \). Ideas here should include recognizing that it is a division of two fractions.

Step 2: Simplify Using Division

The division of two fractions \( \frac{\frac{a}{b}}{\frac{c}{d}} \) can be converted into multiplication by taking the reciprocal of the second fraction. Thus, \[ \frac{\frac{t}{x^{2}-y^{2}}}{\frac{t}{x+y}} = \frac{t}{x^{2}-y^{2}} \times \frac{x+y}{t}. \]

Step 3: Cancel Common Terms

Since we have \( t \) in both the numerator and the denominator, we can cancel out \( t \) from both, resulting in \[ \frac{x+y}{x^{2}-y^{2}}. \]

Step 4: Recognize Difference of Squares

Observe that the expression in the denominator \( x^{2} - y^{2} \) is a difference of squares. This can be rewritten as \( (x-y)(x+y) \). Thus, the complex fraction becomes \[ \frac{x+y}{(x-y)(x+y)}. \]

Step 5: Further Simplification

Notice that \( x+y \) is both a factor of the numerator and the denominator. By canceling \( x+y \) from both, we simplify the fraction to \( \frac{1}{x-y} \) provided that \( x+y eq 0 \).

Key Concepts

Simplifying Complex Fractions

Complex fractions can seem tricky, but breaking them down step by step makes them easier to handle. A complex fraction is a fraction where the numerator or the denominator (or both) contains a fraction. In the exercise provided, the complex fraction is:

• Numerator: \( \frac{t}{x^{2}-y^{2}} \)
• Denominator: \( \frac{t}{x+y} \)

Just remember, simplifying complex fractions often involves performing a fraction division, or turning the problem into a multiplication by using the reciprocal of the denominator fraction. When you encounter a problem like this, keep calm and convert the division into multiplication. This step simplifies even the most daunting complex fractions, making them much more manageable!

Difference of Squares

Recognizing patterns in algebraic expressions, such as the difference of squares, is a great tool for simplifying fractions quickly. In our case, the expression \( x^2 - y^2 \) appears in the denominator and it's a classic example of a difference of squares. This can be factored into:

• \( x^2 - y^2 = (x-y)(x+y) \)

This factoring transforms the fraction into a simpler form and allows for further cancellation. It's important to note that the difference of squares is specific to subtraction and involves perfect squares. Anytime you see something like \( a^2 - b^2 \), think: difference of squares. Factoring these gives you pairs of binomials, which often leads to cancellations when simplifying expressions.

Fraction Division

Fraction division is one of the underlying methods used to simplify complex fractions. It's based on the idea that dividing by a fraction is the same as multiplying by its reciprocal. For the provided exercise, the division of two fractions

• \( \frac{\frac{a}{b}}{\frac{c}{d}} \)

is reimagined as:

• \( \frac{a}{b} \times \frac{d}{c} \)

This transformation makes the fractions much easier to manage, especially when the terms can be cancelled. In the solution, after rewriting the complex fraction as a multiplication, we see that the \( t \) in the numerator can be cancelled with the \( t \) in the denominator, leading us to a simpler expression.
Fraction division essentially relies on flipping the divisor (the second fraction) and multiplying, which helps in understanding and simplifying complex mathematical expressions.