Problem 48 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{x}{7}-\frac{7}{x}}{\frac{1}{7}+\frac{1}{x}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( x - 7 \).

Step 1: Combine the Fractions in the Numerator

The numerator of the complex fraction is \( \frac{x}{7} - \frac{7}{x} \). To combine these fractions, find a common denominator, which is \( 7x \). This yields: \[ \frac{x^2 - 49}{7x} \].

Step 2: Combine the Fractions in the Denominator

The denominator is \( \frac{1}{7} + \frac{1}{x} \). Again, find a common denominator, which is \( 7x \). This gives: \[ \frac{x + 7}{7x} \].

Step 3: Express the Complex Fraction as a Simple Division

The original complex fraction can be expressed as a division: \[ \frac{\frac{x^2 - 49}{7x}}{\frac{x + 7}{7x}} \].

Step 4: Simplify by Multiplying by Reciprocal

Multiply the numerator by the reciprocal of the denominator: \[ \frac{x^2 - 49}{7x} \times \frac{7x}{x + 7} \]. The \( 7x \) terms will cancel out: \[ \frac{x^2 - 49}{x + 7} \].

Step 5: Factor and Simplify Further

Recognize \( x^2 - 49 \) as a difference of squares: \((x - 7)(x + 7)\). Substitute this to get: \[ \frac{(x - 7)(x + 7)}{x + 7} \]. Cancel \( x + 7 \) in the numerator and denominator: \[ x - 7 \]. Hence, the simplified form is \( x - 7 \).

Key Concepts

Understanding the Common Denominator

In order to simplify complex fractions like the one given, finding a common denominator is often the first step. Let’s discuss the concept further. A common denominator is a shared multiple of the denominators of two or more fractions. When fractions have the same denominator, they can be easily added or subtracted. This is useful for combining fractions within complex fractions.Here’s how you do it:

• Identify the denominators in your fractions. For example, in the numerator, the fractions are \( \frac{x}{7} \) and \( \frac{7}{x} \).
• Find the Least Common Denominator (LCD). The LCD should be a multiple of all denominators involved, in this case, it’s \( 7x \).
• Adjust each fraction to have this common denominator. Multiply the numerator and denominator of each fraction so they can be expressed with the LCD.

Once your fractions share a common denominator, you can continue with combining them. Remembering these steps helps to simplify the process of dealing with fractions under a single common umbrella.

Difference of Squares

When working with expressions like \( x^2 - 49 \), recognizing it as a difference of squares can simplify your work. The difference of squares is a specific algebraic pattern, where the expression is structured as \( a^2 - b^2 \).This pattern factors neatly into:

• \( (a - b)(a + b) \)

So, how does that work?

• First, identify what \( a \) and \( b \) are. In our problem, \( a = x \) and \( b = 7 \), because \( x^2 - 7^2 \) can be rewritten using the difference of squares.
• Then, factor it as \( (x - 7)(x + 7) \).

This factoring trick can drastically simplify complex fractions, allowing you to cancel common terms in the numerator and denominator. Understanding and recognizing this pattern is essential, especially for expressions that complicate calculations without this simplification.

Factoring Polynomials

Factoring polynomials is a core skill in algebra that helps simplify expressions, solve equations, and even simplify complex fractions. When we talk about factoring polynomials like \( x^2 - 49 \), it involves breaking down the expression into simpler, multiplied terms.Here's a deeper look at factoring:

• Identify any common factors. In simpler expressions, this can be as straightforward as removing a common multiplication term, but here we use special identities like difference of squares.
• Apply known identities. Our problem used the difference of squares formula, which is just one way of factoring.
  • Others include factoring trinomials into binomials or recognizing perfect square binomials.
• Look for patterns. Polynomials often follow predictable patterns, so with practice, identifying the right method becomes second nature.

By factoring \( x^2 - 49 \) into \( (x - 7)(x + 7) \), you not only simplify the fraction but also make calculations much less complex. Mastery of this technique aids in not just simplifying complex fractions, but in solving a wide array of algebra problems efficiently.