Problem 64 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(\frac{x^2 + x + 1}{x+1}\)

Step 1: Simplifying the Denominator

The original expression is \(1+\frac{x}{1+\frac{1}{x}}\). Focus on the inner-most fraction \(\frac{1}{x}\). Thus, the denominator \(1 + \frac{1}{x}\) can be rewritten as a single fraction with \(x\) as the common denominator: \(\frac{x}{x} + \frac{1}{x} = \frac{x+1}{x}\).

Step 2: Substituting the Simplified Denominator

Replace \(1 + \frac{1}{x}\) in the original expression with \(\frac{x+1}{x}\). The expression now becomes \(1+\frac{x}{\frac{x+1}{x}}\).

Step 3: Simplifying the Complex Fraction

The expression \(\frac{x}{\frac{x+1}{x}}\) can be rewritten as \(x \cdot \frac{x}{x+1}\) by multiplying by the reciprocal of the denominator. This simplifies to \(\frac{x^2}{x+1}\).

Step 4: Adding the Whole Number

Now, return to the entire expression: \(1 + \frac{x^2}{x+1}\). Write \(1\) as \(\frac{x+1}{x+1}\) to combine the terms over a common denominator: \(\frac{x+1}{x+1} + \frac{x^2}{x+1}= \frac{x^2 + x + 1}{x+1}\).

Step 5: Simplified Expression

The expression is now simplified as \(\frac{x^2 + x + 1}{x+1}\).

Key Concepts

Algebraic Expressions

Algebraic expressions form the cornerstone of algebra and consist of variables, numbers, and the operations that combine them, such as addition, subtraction, multiplication, and division. In our example, an expression like \(1+\frac{x}{1+\frac{1}{x}}\) is constructed using these components.

Variables, represented by letters like \(x\), are placeholders for unknown values, which can change depending on the situation. Numbers within algebraic expressions can be constants or coefficients, which give the expression its numerical balance.

• Constants are fixed values, such as numbers like 1.
• Coefficients, like the implicit 1 in front of \(x\), multiply the variables.

Operations like addition \((+)\) and division \((/)\) guide how we interact with these numbers and variables within the expression. Understanding these basics helps in manipulating expressions for simplification or evaluation, as seen in our multi-layered expression. Breaking down a complex expression involves thoughtfully examining its structure and tackling each component layer by layer.

Rational Expressions

Rational expressions are a type of algebraic expression characterized by the presence of a fraction where both the numerator and the denominator are themselves expressions. In algebra, a rational expression might look like \(\frac{x}{1+\frac{1}{x}}\). Here, it's about recognizing the layers as fractions within fractions, which can make problems look much more complicated than they are.

Rational expressions need careful handling, especially:

• Identifying common denominators: As seen in the transformation of \(1+\frac{1}{x}\) into \(\frac{x+1}{x}\).
• Utilizing properties of fractions: Such as converting division of fractions into multiplication by reciprocals, exemplified in the expression \(\frac{x}{\frac{x+1}{x}}\) becoming \(x \cdot \frac{x}{x+1}\).

Approaching rational expressions involves methodically reducing them to simpler forms to clarify their values or meanings, often crucial in solving equations and understanding complex problems in mathematics.

Mathematical Simplification

Mathematical simplification is the process of transforming an expression into its simplest or most manageable form. This is a fundamental skill in mathematics that makes formulas and expressions more understandable and easier to work with.

Simplification can involve reducing the complexity of terms without changing their value. This might include:

• Combining like terms, as seen in final expressions: \(\frac{x^2 + x + 1}{x+1}\).
• Factoring or expanding expressions as needed.

In simplifying the given expression, we see various techniques at play, including finding a common denominator for combination and transforming division operations into multiplication using reciprocals. Successfully simplifying expressions requires not just technical skill but also a strategic vision to anticipate how each transformation will bring us closer to the simplest form possible, such as \(1+\frac{x}{1+\frac{1}{x}}\) reducing to \(\frac{x^2 + x + 1}{x+1}\). This process is not just about performing operations but really understanding the purpose and power of algebraic manipulation.