Problem 72 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{5}{b^{2}}, \frac{4}{b^{3}} $$

Step-by-Step Solution

Answer

Question: Convert the following rational expressions to have the same denominators: $$ \frac{5}{b^2}, \frac{4}{b^3} $$

Step 1: Identify the Current Denominators

The provided rational expressions have the following denominators: $$ \frac{5}{b^2}, \frac{4}{b^3} $$ The denominators are \(b^2\) and \(b^3\).

Step 2: Find the Least Common Multiple (LCM)

The LCM of the denominators is the smallest power of 'b' that is a multiple of both denominators. In this case, the LCM is \(b^3\) because both \(b^2\) and \(b^3\) are factors of \(b^3\).

Step 3: Rewrite the Rational Expressions with the LCM

Now, we need to rewrite each expression with the new common denominator, \(b^3\). For the first expression: $$ \frac{5}{b^2} \cdot \frac{b}{b} = \frac{5b}{b^3} $$ For the second expression: $$ \frac{4}{b^3} = \frac{4}{b^3} $$

Step 4: Present the Final Answer

The rational expressions with the same denominator are: $$ \frac{5b}{b^3}, \frac{4}{b^3} $$

Key Concepts

Least Common Multiple

To work with rational expressions effectively, understanding the concept of the least common multiple (LCM) is essential. The LCM is a mathematical value that represents the smallest number or expression that is a common multiple of two or more numbers or expressions. When dealing with denominators, finding the LCM allows us to combine fractions or rational expressions by ensuring they all have the same base for arithmetic operations. To find the LCM of algebraic expressions, look for the highest power of each variable appearing in any of the expressions. For instance, if you have denominators like \(b^2\) and \(b^3\), the LCM is \(b^3\), because it contains each variable at its highest power.

Denominator

In the context of fractions or rational expressions, the denominator is the bottom part of the fraction that indicates into how many equal parts the whole is divided. It provides the basis for operations like addition, subtraction, and comparison. When we deal with algebraic fractions, the denominators can be expressions involving one or more variables, such as \(b^2\) and \(b^3\). Having a common denominator, as found using the LCM, is crucial for combining fractions and ensuring precise calculations. Without a common denominator, it's challenging to perform direct arithmetic operations between different fractions.

Algebraic Fractions

Algebraic fractions are similar to numeric fractions, but they involve variables in their numerators and/or denominators. These fractions follow the same principles as numerical fractions, but they require careful handling of algebraic expressions. For example, in the exercise, the terms are \(\frac{5}{b^2}\) and \(\frac{4}{b^3}\). Here, the numerators are numerical (5 and 4), while the denominators are algebraic expressions involving the variable \(b\). Working with algebraic fractions involves finding common denominators and simplifying the expressions without altering their values, ensuring correct mathematical operations.

Simplifying Expressions

Simplifying expressions is a key part of working with rational expressions. It involves rewriting expressions in a simpler or more efficient form without changing their value. This can make calculations easier and solutions more understandable. In our exercise, simplifying took place when we multiplied the first expression by \(\frac{b}{b}\) to make its denominator \(b^3\). The original fraction \(\frac{5}{b^2}\) becomes \(\frac{5b}{b^3}\) after multiplication, thus enabling consistent denominators.Simplifying expressions involves reducing terms, cancelling common factors, and combining like terms, which is crucial for clearer and more efficient calculations.