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Chapter 9: Problem 23

Simplify each expression. $$ \frac{m-3}{m(2 m-6)} $$

Short Answer

Expert verified
\( \frac{1}{2m} \)

Step by step solution

01

Identify Common Factors

Look at the expression and identify any common factors in the numerator and the denominator. The given expression is \( \frac{m-3}{m(2m-6)} \).
02

Factor the Denominator

Factor the denominator, \( m(2m-6) \). Notice that \( 2m - 6 \) can be factored as \( 2(m-3) \). Thus, the denominator becomes \( m \times 2(m-3) \), which simplifies to \( 2m(m-3) \).
03

Rewrite the Expression

Substituting the factored form into the expression, we get: \[ \frac{m-3}{2m(m-3)} \].
04

Simplify the Expression

Cancel out the common factor \( m-3 \) in the numerator and the denominator.The simplified form of the expression becomes: \[ \frac{1}{2m} \].

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Key Concepts

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

Factoring Expressions
Factoring is a key process in algebra where you break down an expression into a product of simpler expressions, also known as 'factors'. For example, consider the expression \(2m-6\). Notice that both terms have a common factor of 2. Hence, you can rewrite it as \(2(m-3)\). The process involves:
  • Identifying the greatest common factor (GCF) of the terms
  • Writing the expression in terms of this GCF
Factoring helps in simplifying complex algebraic expressions and making them easier to work with.
Canceling Common Factors
Once you have factored an expression, you might find that the numerator and denominator share common factors. Canceling these common factors simplifies the expression. For instance, in the fraction \( \frac{m-3}{2m(m-3)} \), the \((m-3)\) term appears in both the numerator and the denominator. You can 'cancel out' this term because dividing by a term and multiplying by the same term are inverse operations. After canceling \((m-3)\), the fraction becomes: \( \frac{1}{2m} \). Remember:
  • Factor first before canceling common terms
  • Check that you are canceling terms, not individual factors within a term
Canceling common factors simplifies the rational expression and makes it more manageable.
Rational Expressions
A rational expression is simply a fraction where the numerator and denominator are polynomials. In our example, \( \frac{m-3}{m(2m-6)} \) is a rational expression. Working with rational expressions involves:
  • Simplifying the expression by factoring and canceling common factors
  • Ensuring that the expression remains mathematically equivalent throughout the simplification process
  • Understanding that the domain of the rational expression excludes any values that make the denominator zero
For our example, after simplification, the rational expression is \( \frac{1}{2m} \), excluding \( m = 0 \) and \( m = 3 \) from its domain to avoid division by zero.

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