Problem 42 - Algebra 1

Question

Statement Explanation 1\. \(\frac{a c}{b c}=\frac{a}{b} \cdot \frac{?}{?}\) 1\. Apply the rule for multiplying rational expressions. 2\. \(\underline{?}=\frac{a}{b} \cdot \underline{?}\) 2\. Any nonzero number divided by itself is 1. 3\. \(\underline{?}=\frac{a}{b}\) 3\. Any nonzero number multiplied by 1 is itself. Copy and complete the proof to show why you can divide out common factors.

Step-by-Step Solution

Answer

In the rational expression \(\frac{a c}{b c}=\frac{a}{b} \cdot \frac{c}{c}\), the \(c\) in the numerator and denominator of the right side cancel each other out, as any nonzero number divided by itself is 1. Multiplying \(\frac{a}{b}\) by 1 gives the result as \(\frac{a}{b}\). Hence, we have proven that in a rational expression, we can always divide out common factors.

Step 1: Filling the first gap

We apply the rule for multiplying rational expressions, so \(\frac{a c}{b c}=\frac{a}{b} \cdot \frac{c}{c}\)

Step 2: Filling the second gap

Any nonzero number divided by itself is 1. Therefore, \(\frac{a}{b} \cdot \frac{c}{c}=\frac{a}{b} \cdot 1\)

Step 3: Filling the third gap

Any nonzero number multiplied by 1 retains its original value. Therefore, \(\frac{a}{b} \cdot 1=\frac{a}{b}\)

Key Concepts

Rational expressions

Understanding rational expressions is crucial when working with algebraic fractions. A rational expression is essentially a fraction in which both the numerator and the denominator are polynomials. For instance, \( \frac{a}{b} \) is a rational expression provided \( b \) is not zero. In our exercise, \( \frac{a c}{b c} \) is a rational expression where \( ac \) and \( bc \) are polynomials and \( bc \) is not zero.

When multiplying rational expressions like \( \frac{a c}{b c} \) with \( \frac{c}{c} \) as we did in the exercise, we leverage the property that multiplication of fractions is essentially the multiplication of their numerators and denominators, respectively. It's vital to simplify the expression, whenever possible, to its lowest terms by dividing out common factors from the numerator and the denominator – this is the essence of the solution to our problem.

Common factors in algebra

The identification and elimination of common factors is a powerful tool in algebra for simplifying expressions and solving equations. Common factors are the numbers or variables that appear in both the numerator and the denominator of a fraction. In the context of our exercise, \( c \) was a common factor in both the numerator \( ac \) and denominator \( bc \) of the rational expression \( \frac{a c}{b c} \).

Recognizing the common factor \( c \) allows us to divide both the numerator and the denominator by \( c \) to simplify the expression – reflecting the fundamental property of division that states a number divided by itself is equal to one. Simplifying expressions by removing common factors reduces complexity and often leads to more direct and easier-to-understand solutions. This method can be used across polynomial and rational expressions alike, making it a cornerstone of algebra.

Properties of division

Division is an essential operation in algebra with properties that significantly influence the way we manipulate expressions. One key property is that any nonzero number divided by itself equals one. This property was used in the exercise to transform \( \frac{c}{c} \) into 1. Such a transformation is crucial as it simplifies the multiplication by reducing it to \( \frac{a}{b} \cdot 1 \).

Another important property of division is that multiplying any number by 1 leaves the original number unchanged. We leveraged this property in the final step of the exercise when we multiplied \( \frac{a}{b} \cdot 1 \) to get \( \frac{a}{b} \) as the result. Understanding these properties allows students not only to perform operations correctly but also to recognize opportunities to simplify and solve algebraic expressions more efficiently.