Problem 48 - Introductory Algebra for College Students

Question

Simplify complex rational expression. \(\frac{1}{1+\frac{1}{1+\frac{1}{2}}}\)

Step-by-Step Solution

Answer

The simplified form of the complex rational expression \(\frac{1}{1+\frac{1}{1+\frac{1}{2}}}\) is approximately \(0.60\)

Step 1: Simplify the innermost fraction

First, we address the fraction that is deepest within the expression, which is \(1/2\). Since \(1/2=0.5\), the expression simplifies to: \(1/(1+0.5)\) or \(1/1.5\)

Step 2: Simplify the next layer fraction

Next, we simplify the next fraction \(1/(1+1/1.5)\), but first we must resolve the expression inside the parentheses. To do this, simplify \(1+1/1.5\) by first dividing \(1/1.5 = 0.667\) and then adding that result to \(1\), which gives us about \(1.667\). Therefore, the expression simplifies to: \(1/1.667\)

Step 3: Simplify the outermost layer

Finally, to get our answer, divide \(1\) by \(1.667\) to give us about \(0.60\) as the simplest form of given complex rational expression.

Key Concepts

Understanding Fractions

Fractions are a vital part of mathematics, representing division of a whole. Understanding them is crucial in simplifying complex rational expressions. A fraction has two parts: a numerator (the top number) and a denominator (the bottom number).

• The numerator tells how many parts we have.
• The denominator indicates into how many parts the whole is divided.

To simplify fractions, we find the greatest common divisor of the numerator and the denominator and divide both by it. This process doesn't change the fraction's value, just its form.
When dealing with nested fractions like in our exercise, it’s important to start simplifying from the innermost fraction. By doing so, you gradually work your way outward, turning a complex expression into an easier, simpler form.

Algebra Basics Overview

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It often involves finding the value of an unknown variable.
Here are some basics that are crucial for simplifying complex expressions:

• Order of Operations: Always follow the PEMDAS/BODMAS rule - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Variables and Constants: Variables are big in algebra; they represent unknown values. Constants are known values.
• Simplification Techniques: This involves breaking down expressions into smaller parts and resolving them one step at a time. Simplifying inside brackets first is critical.

In the exercise, using algebraic principles helps us resolve nested fractions in an orderly manner. By simplifying each step diligently, we arrive at the final simplified expression.

Step-by-Step Solutions Explained

Breaking down expressions step-by-step helps clarify each part of the problem and makes the overall task more manageable. In our exercise, we follow a methodical approach, which consists of:

• Step 1: Innermost Fraction Simplification: We tackled the smallest fraction first, making our job easier as it reduced complexity from the inside out.
• Step 2: Middle Layer Simplification: After dealing with the innermost fraction, the next level was simplified by concentrating on calculations within the parentheses. This step prepares the expression for final simplification.
• Step 3: Outer Layer Simplification: Simplifying the final expression involved computing the outermost operation, resulting in an easy-to-understand answer.

This structured approach helps in understanding not only the solution to the problem but the process and logic behind it. It builds a solid foundation for tackling more complex algebraic expressions in the future. Implementing such step-by-step solutions ensures comprehensive comprehension and mastery of simplifying complex expressions.