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Chapter 10: Problem 3

In Exercises \(1-4,\) replace \(k\) by \(k+1\) in each expression. $$\frac{k}{k+1}$$

Short Answer

Expert verified
The resulting expression after replacing 'k' by 'k+1' in \(\frac{k}{k+1}\) is \(\frac{k+1}{k+2}\)

Step by step solution

01

Identify the 'k' terms to be replaced

In the given expression \(\frac{k}{k+1}\) both the numerator and denominator contain the variable 'k'. Each of these 'k' terms will be replaced with 'k+1'.
02

Substitution of 'k' with 'k+1'

Replace each 'k' in the expression with 'k+1'. The expression becomes: \(\frac{k+1}{(k+1)+1}\)
03

Simplify the denominator

The denominator in step 2 simplifies to 'k+2'. So the final resulting expression after replacing 'k' by 'k+1' in the original expression \(\frac{k}{k+1}\) is \(\frac{k+1}{k+2}\)

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Key Concepts

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

Substitution in Algebraic Expressions
Substitution is an essential technique in algebra that involves replacing a variable in an expression with another expression or numeral. This process is akin to replacing a component in a machine with an updated or different component to see how the overall system changes. For example, in the expression \( \frac{k}{k+1} \), we are asked to substitute 'k' with 'k+1'. This substitution results in the expression becoming \( \frac{k+1}{(k+1)+1} \).

Substitution can help in various scenarios:
  • It can simplify expressions to a point where they are easier to work with or solve.
  • It allows exploring how changes in one part of an expression affect the whole.
  • It's a fundamental step in solving equations, making substitutions crucial for finding solutions.
Simplification of Algebraic Expressions
Simplification is the process of reducing an expression to its simplest form. This is achieved by performing mathematical operations that combine like terms and reduce complexity without changing the expression's value. In our example, after substituting \( k \) with \( k+1 \), the expression \( \frac{k+1}{(k+1)+1} \) can be simplified.

Specifically, the goal is to make the expression cleaner and more manageable. Our given expression simplifies as follows:
  • The denominator \( (k+1)+1 \) simplifies to \( k+2 \).
  • The entire expression thus becomes \( \frac{k+1}{k+2} \).
Simplification makes the expression easier to read and use, particularly if one needs to perform additional manipulations or solve the expression.
Understanding the Denominator in Fractions
The denominator is a critical part of any fraction, representing the number of equal parts into which the whole is divided. It functions as the 'base' or 'bottom' of the fraction. In the expression \( \frac{k}{k+1} \), the original denominator is \( k+1 \).

Changing the expression involves understanding how modifications to the denominator affect the entire fraction. When we substitute to form \( \frac{k+1}{k+2} \), the denominator changes from \( k+1 \) to \( k+2 \).
  • This new denominator means the fraction represents a different division of a whole, which can influence the fraction's behavior or value.
  • Always simplify where possible as this helps in clearer interpretations and calculations.
Recognizing how the denominator interacts with the numerator is vital when evaluating fractions in algebra and mathematics as a whole.

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