Problem 47 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x}{x+1}\)

Step 1: Identify the innermost fraction

The innermost fraction and hence the first to solve is \(\frac{1}{x}\). It is advisable to deal with this fraction directly but should it present any hurdles, it could be looked at as \(x^{-1}\).\n

Step 2: Simplify the next fraction

The next fraction to simplify is \(1 + \frac{1}{x}\). Evaluating \(\frac{1}{x}\) as \(x^{-1}\) and replacing in the expression we get, \(1 + x^{-1}\). If you would want to have a single denominator, you may think of \(1\) as \(x/x\). This them becomes \(\frac{x+1}{x}\).

Step 3: Simplify the outermost fraction

Now we proceed to the outermost fraction which is \(\frac{1}{1+\frac{1}{x}}\). From Step 2 we know that, \(1 + \frac{1}{x} = \frac{x+1}{x}\). After substitution into the outermost fraction you obtain \(\frac{1}{\frac{x+1}{x}}\). This can be viewed as a division. When we divide by a fraction, we multiply by its reciprocal therefore the simplified expression becomes \(1*\frac{x}{x+1} = \frac{x}{x+1}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, which greatly benefits when you can work with less complex numbers. This is especially true in cases of complex rational expressions like nested fractions, where the goal is to transform them into easier-to-manage single-level fractions. A fraction simplifies by identifying common factors in the numerator and denominator and then reducing them.

• For the provided expression, simplification begins by identifying the innermost fraction, which is \(\frac{1}{x}\).
• By simplifying each component and combining them thoughtfully, one can unravel the expression into its simplest form.

A clear understanding of how to manage and manipulate fractions through methods such as finding a common denominator or factoring can make a significant difference in tackling algebra problems.

Reciprocals in Algebra

Reciprocals are vital in simplifying complex rational expressions. A reciprocal in algebra is the inverse of a number or fraction, effectively flipping it upside down. For example, the reciprocal of \(\frac{1}{x}\) is \(x\). Reciprocals are crucial when dividing fractions as they turn the division into multiplication.

When dealing with complex rational expressions, identifying when and how to use the reciprocal transforms daunting calculations into manageable ones.

• In our exercise, the key step is realizing that dividing by a fraction \(\left(\frac{x+1}{x}\right)\) equates to multiplying by its reciprocal \(\left(\frac{x}{x+1}\right)\).
• Understanding this concept simplifies finalizing the expression to its most manageable form.

Mastering the use of reciprocals can greatly accelerate solving and simplifying fractions in algebra, making them less overwhelming.

Fraction Division

Fraction division often confuses students, yet it follows a simple rule of multiplying by the reciprocal. Whenever we encounter division in rational expressions, we convert it into a multiplication by flipping the second fraction.

Consider how this applies in our original problem:

• After simplifying the inner fraction to \(\frac{x+1}{x}\), we approached the outer fraction.
• We have \(\frac{1}{\left(\frac{x+1}{x}\right)}\). This division can be turned into a multiplication — \(1 \times \frac{x}{x+1}\).

By practicing this approach consistently, simplifying rational expressions becomes second nature. Recognizing opportunities to divide fractions and applying the reciprocal multiplication method will significantly enhance your algebraic proficiency.