/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Perform the indicated operations... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 46

Perform the indicated operations and simplify. Check the solution with a graphing calculator. $$1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}$$

Short Answer

Expert verified
The simplified expression is \( \frac{2x+1}{x+1} \).

Step by step solution

01

Simplify Innermost Fraction

Start with the innermost fraction of the given expression: \( \frac{1}{x} \). We need to work our way outward in the given expression: \[ 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \]
02

Simplify Next Layer

Take the expression from the last step \( \frac{1}{x} \) and substitute it:\[ 1 + \frac{1}{1 + \frac{1}{x}} \_ \]Inside the denominator, calculate \( 1 + \frac{1}{x} \), substituting that into the next layer:\[ \frac{1}{1 + \frac{1}{x}} \_ \]This results in:\[ \frac{x}{x+1} \] as the new expression.
03

Simplify Next Outer Layer

Substitute the simplified expression from the previous step:\[ 1 + \frac{x}{x+1} \] Combine this by expressing \(1\) as a fraction with a common denominator:\[ \frac{x+1}{x+1} + \frac{x}{x+1} \_ \]This results in:\[ \frac{x+1+x}{x+1} = \frac{2x+1}{x+1} \] as the simplified expression.
04

Verify with a Graphing Calculator

Use a graphing calculator to check the expression \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \). Enter it as \( \frac{2x+1}{x+1} \) and ensure that the graph is consistent or use the calculator's fraction simplification to verify the equivalence of the two expressions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Simplification
Fraction simplification involves reducing a complex fraction into its simplest form. This process is essential in algebra to make expressions easier to understand and work with. In our given expression, we have a nested fraction:
  • Start from the innermost fraction, which is \( \frac{1}{x} \).
  • Combine fractions by finding a common denominator.
Working from the inside out is a key strategy here. For example, after dealing with \( \frac{1}{x} \), you combine it with 1, i.e., \( 1 + \frac{1}{x} \), which forms \( \frac{x+1}{x} \).
The substitution method further simplifies this expression layer by layer until it is reduced to its simplest form \( \frac{2x+1}{x+1} \).
By maintaining attention to each nested fraction step-by-step, fraction simplification becomes manageable and neat.
Substitution Method
The substitution method is a powerful tool for simplifying complex algebraic expressions. In this process, you substitute fractions progressively from innermost to outermost layers. For our expression, we:
  • Begin by simplifying \( \frac{1}{x} \).
  • Substitute this into \( 1 + \frac{1}{x} \), obtaining \( \frac{x+1}{x} \).
  • Then, use it in the next expression: \( \frac{1}{1 + \frac{1}{x}} \), simplified to \( \frac{x}{x+1} \).
  • Finally, replace into the larger expression: \( 1 + \frac{x}{x+1} \), which simplifies to \( \frac{2x+1}{x+1} \).
The substitution method allows breaking down a problem into smaller, more manageable parts. By focusing on one piece at a time, you can simplify intricate algebraic expressions with clarity and accuracy.
Verifying Solutions with Technology
Verification using technology, such as graphing calculators, is an excellent strategy to ensure the accuracy of algebraic solutions. In this exercise, a graphing calculator can quickly confirm the correctness of our simplified expression. Here's how you can use a graphing calculator effectively:
  • Input the original expression \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \).
  • Also, input the simplified version \( \frac{2x+1}{x+1} \).
  • Compare both outputs for equivalence, either graphically or numerically.
Technology not only validates algebraic manipulations but also helps to visualize how expressions behave. This method boosts your confidence in the solutions, ensuring they are both correct and efficient.

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