Problem 55 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{a-4+\frac{1}{a}}{-\frac{1}{a}-a+4} $$

Step-by-Step Solution

Answer

\( \frac{a^2 - 4a + 1}{-a^2 + 4a - 1} \)

Step 1: Identify the Inner Fractions

Examine the complex fraction \( \frac{a-4+\frac{1}{a}}{-\frac{1}{a}-a+4} \). Identify that both the numerator and the denominator contain the inner fraction \( \frac{1}{a} \). We will need to address these inner fractions to simplify the entire expression.

Step 2: Create a Common Denominator

In both the numerator \( a - 4 + \frac{1}{a} \) and the denominator \( -\frac{1}{a} - a + 4 \), express each part over a common denominator. Use \( a \) as the common denominator for both. For the numerator, write it as \( \frac{a(a) - 4(a) + 1}{a} \). Similarly, for the denominator, write it as \( \frac{-1 - a(a) + 4a}{a} \).

Step 3: Simplify the Numerator and Denominator

Combine and simplify the expressions you created. For the numerator, simplify to \( \frac{a^2 - 4a + 1}{a} \). For the denominator, simplify to \( \frac{-a^2 + 4a - 1}{a} \). Both expressions now have a common denominator, making the next steps more straightforward.

Step 4: Rewrite the Complex Fraction

Rewrite the original expression using the common denominators: \( \frac{\frac{a^2 - 4a + 1}{a}}{\frac{-a^2 + 4a - 1}{a}} \). Since both the numerator and denominator are fractions with the same denominator, you can divide them directly.

Step 5: Simplify the Overall Fraction

Simplify by canceling the common denominator \( a \) in both the numerator and denominator: \( \frac{a^2 - 4a + 1}{-a^2 + 4a - 1} \). This is the simplified form of the complex fraction.

Key Concepts

Simplification

Simplification is an essential skill in mathematics, especially when dealing with complex fractions. It involves reducing a complicated expression into its simplest form. With complex fractions like \( \frac{a-4+\frac{1}{a}}{-\frac{1}{a}-a+4} \), the simplification process starts by addressing the fractions within the larger fraction.

• Begin by identifying all inner fractions within both the numerator and the denominator. These are smaller fractions that complicate the expression.
• The next step is to eliminate these inner fractions, usually by finding a common denominator for both the numerator and the denominator.
• This helps transform a complex fraction into a more manageable form, eventually allowing for easier simplification by cancellation.

Simplification can make an otherwise daunting expression manageable by reducing it to its simplest form. This not only makes further calculations easier but also aids in the intuitive understanding of the mathematical expression.

Common Denominator

The concept of a common denominator is crucial in simplifying complex fractions. When you have fractions within the numerator and denominator, the process becomes much smoother by first rewriting each part over a common denominator. For our example, consider the expression \( \frac{a-4+\frac{1}{a}}{-\frac{1}{a}-a+4} \).

• The presence of \( \frac{1}{a} \) in both the numerator and denominator indicates that \( a \) is the natural choice for a common denominator.
• Rewrite each term in both the numerator and denominator such that they all share this common denominator. The numerator becomes \( \frac{a(a) - 4(a) + 1}{a} \) and the denominator turns into \( \frac{-1 - a(a) + 4a}{a} \).

Using a common denominator not only simplifies the expression process but also facilitates direct comparison and combination of the terms involved, thereby streamlining the simplification steps.

Algebraic Expressions

Algebraic expressions are a fundamental component of solving complex fractions. Each term in expressions like \( \frac{a^2 - 4a + 1}{-a^2 + 4a - 1} \) involves variables and constants combined using arithmetic operations.

• Understanding algebraic expressions requires recognizing operations such as addition, subtraction, multiplication, or division being performed on variables and numbers.
• In complex fractions, evaluate both the numerator and the denominator separately. Simplifying these parts requires combining like terms and reducing the expression to its simplest form.
• With these steps, you transform a cumbersome complex fraction into a clearer form which reflects the relationship between the variables involved.

Working with algebraic expressions within complex fractions enhances analytical skills and requires a solid understanding not only of algebraic manipulation but also of the relationships between the components of a given mathematical expression.