Problem 47 - Prealgebra and Introductory Algebra

Question

Write each expression in terms of the LCM of the denominators. $$b ; \frac{5}{b}$$

Step-by-Step Solution

Answer

The expression \(b\) can be written as \(\frac{b^2}{b}\) and the expression \(\frac{5}{b}\) stays as it is when both are written in terms of the LCM of their denominators.

Step 1: Identify the expressions and their denominators

The expressions are \(b\) and \(\frac{5}{b}\). Write \(b\) as \(\frac{b}{1}\). So, the denominators are \(1\) and \(b\).

Step 2: Find the least common multiple of the denominators

The least common multiple (LCM) of \(1\) and \(b\) is \(b\).

Step 3: Rewrite each term with the LCM as the denominator

The term \(b\) can be written as \(\frac{b}{1}\). So, we multiply the numerator and denominator of \(\frac{b}{1}\) by \(b\) to get \(\frac{b^2}{b}\). For the term \(\frac{5}{b}\), the denominator is already the LCM (which is \(b\)), so no changes are needed. We now have the terms as \(\frac{b^2}{b}\) and \(\frac{5}{b}\).

Key Concepts

Understanding Rational Expressions

Rational expressions are mathematical phrases that are structured like fractions. However, instead of just numbers in the numerator and denominator, they may contain variables, too. Rational expressions often look like this: \( \frac{5}{x+2} \), where \(5\) is the numerator and \(x+2\) is the denominator.
They are called "rational" because they express the ratio of two quantities. A critical point to remember about rational expressions is that the denominator cannot be zero. This is because division by zero is undefined.

• These expressions allow us to understand more complex fractions involving variables.
• They are similar to numerical fractions and follow similar rules.

Simplifying rational expressions often involves factoring and finding common denominators, especially when adding or comparing them. Understanding these elements allows you to work with these expressions easily and avoid errors in calculations.

The Role of Denominators

The denominator is the number or expression below the line in a fraction or rational expression. It signifies the total number of parts that make up a whole.
For instance, in the fraction \( \frac{3}{4} \), \(4\) is the denominator. It tells us that the whole is divided into four equal parts.

• In rational expressions, denominators can consist of both numbers and variables.
• They can never be zero, as this would make the expression undefined.

Finding the Least Common Multiple (LCM) of denominators is crucial when adding, subtracting, or comparing rational expressions. The LCM is the smallest number or expression that each denominator divides into evenly, allowing expressions to be written with a common denominator.

Working with Fractions

Fractions are numbers that represent a part of a whole. They are written with a numerator above a line and a denominator below it. There are several operations you can perform with fractions, such as addition, subtraction, multiplication, and division.
To add or subtract fractions, you need a common denominator. This is where the concept of the Least Common Multiple (LCM) becomes useful.

• For example, adding \( \frac{1}{2} \) and \( \frac{1}{3} \): here, the LCM of the denominators \(2\) and \(3\) is \(6\), so you convert both to have \(6\) as their denominator.
• Once they have common denominators, you simply add the numerators, leaving the denominator unchanged.

Understanding fractions is essential to tackling more complex mathematical concepts, as it's foundational in learning about rational expressions.