Problem 47 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{8}{x}+\frac{x}{8}}{\frac{x}{8}-\frac{8}{x}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{x^2 + 64}{(x-8)(x+8)} \).

Step 1: Combine the Fractions in the Numerator

In the numerator, we have the expression \( \frac{8}{x} + \frac{x}{8} \). To combine these, find a common denominator. The common denominator for \( x \) and \( 8 \) is \( 8x \). Thus, rewrite the fractions as follows: \( \frac{8 \cdot 8}{x \cdot 8} + \frac{x \cdot x}{8 \cdot x} = \frac{64}{8x} + \frac{x^2}{8x} \). Combine them: \( \frac{64 + x^2}{8x} \).

Step 2: Combine the Fractions in the Denominator

Similarly, in the denominator, we have \( \frac{x}{8} - \frac{8}{x} \). Use the common denominator \( 8x \), and rewrite the fractions as \( \frac{x \cdot x}{8 \cdot x} - \frac{8 \cdot 8}{x \cdot 8} = \frac{x^2}{8x} - \frac{64}{8x} \). Combine them: \( \frac{x^2 - 64}{8x} \).

Step 3: Divide the Numerator by the Denominator

The complex fraction becomes \( \frac{\frac{64 + x^2}{8x}}{\frac{x^2 - 64}{8x}} \). To simplify, divide the two fractions by multiplying by the reciprocal: \( \frac{64 + x^2}{8x} \times \frac{8x}{x^2 - 64} \).

Step 4: Simplify the Resulting Expression

Notice \( \frac{8x}{8x} = 1 \), simplifying it leads to \( \frac{x^2 + 64}{x^2 - 64} \). We observe that numerator cannot be further factorized directly, whereas the denominator is a difference of squares: \( x^2 - 64 = (x-8)(x+8) \).

Step 5: Conclusion

The simplified form of the complex fraction is \( \frac{x^2 + 64}{(x-8)(x+8)} \), as no common factors exist that can be cancelled out further.

Key Concepts

Algebra

When dealing with algebra, it is crucial to understand how to manipulate equations and expressions. Algebra often involves variables, which are symbols used to represent unknown numbers. You can arrange these symbols along with numbers to form expressions and equations. In the context of complex fractions, algebra is used to find equal values and simplify expressions.

One basic concept in algebra is the idea of equivalent expressions. Expressions are equivalent if they have the same value. For complex fractions, finding a common denominator allows us to combine separate fractional expressions into one, simplifying the overall calculation.

• Variables: Represent unknowns or any number in expressions (e.g., \(x\))
• Expressions: Combination of numbers and variables (e.g., \(\frac{8}{x} + \frac{x}{8}\))
• Equations: A statement that two expressions are equal, often solved to find the value of the unknown

Understanding these concepts can help in manipulating and simplifying complex fractions by leveraging the properties and rules of algebra.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form. This means rewriting them so that the numerator and the denominator have no common factors other than 1.
In the case of complex fractions, you work with fractions within a fraction. The simplification process involves several steps:

• Finding a common denominator for fractions in the numerator or denominator to combine them into a single fraction.
• Recognizing patterns such as the difference of squares, which aid in factorizing expressions.
• Simplifying by canceling out common factors present in the numerator and the denominator.
  

Let's break down the steps: First, convert the fractional components to common denominators. Then combine them into one fraction. Finally, if possible, factorize and cancel common factors to reach the simplest form.
Through these strategies, complex fractions can become far more manageable and their values easier to understand.

Mathematical Expressions

Mathematical expressions are foundational elements of math. They consist of numbers, variables, and operational symbols that together express a quantity or a relationship. In dealing with complex fractions, being able to recognize and manipulate these expressions is key.
The key to simplifying complex fractions lies in tidy manipulation of these expressions to achieve a simpler form. For instance, leveraging the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\), allows easy factorization and simplification. Identifying such patterns can greatly aid in the expression manipulation process.
Moreover, expressions also require recognizing and using reciprocal operations. When dividing a fraction by another, consider multiplying by the reciprocal instead, because dividing by a fraction is the same as multiplying by its inverse.
In sum, understanding how to effectively work with mathematical expressions will improve your abilities to tackle everything from basic calculations to more complex algebraic problems, such as those involving complex fractions.