Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ 1-\frac{1}{1-\frac{1}{1-x}} $$

Short answer

\( \frac{1}{x} \)

Step-by-step solution

Simplify the innermost fraction

Start by simplifying the innermost fraction \ \( \frac{1}{1-x} \). This remains as is because it is already simplified. So we have \ \[ 1 - \frac{1}{1 - \frac{1}{1-x}} \]

Substitute the innermost fraction

Next, substitute \ \( \frac{1}{1-x} \) into the equation. The expression inside the larger fraction now has the form \ \[ 1 - \frac{1}{1 - \frac{1}{1-x}} = 1 - \frac{1}{1 - \left( \frac{1}{1-x} \right)} \]

Simplify the denominator of the large fraction

Now simplify the denominator of the large fraction \ \(1 - \frac{1}{1-x} \). To do this, find a common denominator, which is \ \(1-x \): \ \[ 1 - \frac{1}{1-x} = \frac{(1-x) - 1}{1-x} = \frac{1-x-1}{1-x} = \frac{-x}{1-x} \]

Simplify the large fraction

Now we have: \ \[1 - \frac{1}{\frac{-x}{1-x}} = 1 - \frac{1-x}{-x} = 1 + \frac{1-x}{x} \]. Break this into two separate fractions: \ \[ 1 + \frac{1}{x} - \frac{x}{x} = 1 + \frac{1}{x} - 1 \]

Combine like terms

Combine the like terms to simplify the expression: \ \[ 1 - 1 + \frac{1}{x} = \frac{1}{x} \]

Write the answer in factored form

The expression \ \( \frac{1}{x} \) is already in its simplest and factored form.

Key concepts

Fraction Simplification

Simplifying fractions is a core skill in algebra. Let's break it down step by step.
We often deal with fractions within fractions, known as complex fractions. To simplify, we start from the innermost fraction and work our way out. For instance, in the original exercise, the innermost fraction is \( \frac{1}{1-x} \). It can't be simplified further at this point.
Hence, the goal is to simplify the larger fraction that contains this innermost fraction.
Remember, a simplified fraction is one where the numerator and denominator have no common factors other than 1.

• Identify the innermost fractions first.
• Simplify them step-by-step.
• Combine and reduce where possible.

This methodical approach ensures accurate simplification.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules.
In the exercise, we deal with nested fractions that need careful manipulation.
Here’s a trick: Focus on the denominator first. We had \(1 - \frac{1}{1-x} \), and we needed a common denominator to combine them.
The common denominator is \( 1-x \): \[ 1 - \frac{1}{1-x} = \frac{1(1-x) - 1}{1-x} = \frac{1-x-1}{1-x} = \frac{-x}{1-x} \] Performing these steps allows you to simplify fractions accurately.
When multiplying or dividing fractions, multiply numerators together and denominators together.
Always look out for factors that can be canceled to simplify further.
Mastering these basics will make dealing with rational expressions much more manageable.

Factored Form

Factored form means expressing an algebraic expression as a product of its factors.
It simplifies expressions and solves equations. In our example, reaching the final answer \( \frac{1}{x} \) means we've put it in the simplest form.
Each step simplified the expression to it's factorized form. Rewriting fractions in factored form can reveal simplifications and solutions more easily.
Useful tips for factoring:

• Identify common factors.
• Apply the distributive property: \( a(b + c) = ab + ac \).
• Recognize patterns such as differences of squares: \( a^2 - b^2 = (a+b)(a-b) \).

Always aim to leave your final answer in this form for clarity and precision.