Problem 74 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{9}{x^{2}}, \frac{1}{4 x} $$

Step-by-Step Solution

Answer

Question: Convert the following rational expressions to have the same denominator: \(\frac{9}{x^{2}}\) and \(\frac{1}{4x}\). Answer: The rational expressions with the same denominator are \(\frac{36}{4x^{2}}\) and \(\frac{x}{4x^{2}}\).

Step 1: Determine the Least Common Multiple (LCM) of the denominators

The given rational expressions contain the denominators \(x^{2}\) and \(4x\). To find the LCM of these two denominators, first, look for the highest power of each variable: - For \(x\), the highest power is \(x^{2}\). Next, consider the numerical part, which means finding the LCM of 1 and 4. The LCM of 1 and 4 is 4. Thus, the LCM of the denominators is \(4x^{2}\).

Step 2: Convert the rational expressions to have the LCM as the denominator

Now that we have the LCM, we need to rewrite each fraction with the denominator of \(4x^{2}\): 1. For the first rational expression \(\frac{9}{x^{2}}\), multiply both the numerator and the denominator by \(\frac{4}{4}\) to get the denominator \(4x^{2}\) $$ \frac{9}{x^{2}} \cdot \frac{4}{4} = \frac{9 \cdot 4}{x^{2} \cdot 4} = \frac{36}{4x^{2}} $$ 2. For the second rational expression \(\frac{1}{4x}\), multiply both the numerator and the denominator by \(\frac{x}{x}\) to get the denominator \(4x^{2}\) $$ \frac{1}{4x} \cdot \frac{x}{x}=\frac{1 \cdot x}{4x \cdot x} = \frac{x}{4x^{2}} $$

Step 3: Present the final rational expressions with the same denominator

The given rational expressions have now been converted to rational expressions with the same denominator of \(4x^{2}\). The converted rational expressions are: $$ \frac{36}{4x^{2}}, \frac{x}{4x^{2}} $$

Key Concepts

Least Common Multiple

Understanding the Least Common Multiple (LCM) is crucial when dealing with rational expressions, especially when you need to work with multiple fractions. The LCM is the smallest expression (or number, when dealing with integers) that is divisible by each of the denominators you're working with.

• For algebraic fractions, the LCM is determined by taking the highest power of each variable present in the denominators.
• Additionally, consider the coefficients for numerical parts, and find the LCM of these numbers as usual.

For example, if your denominators are \(x^2\) and \(4x\), the LCM would involve the highest power of \(x\), which is \(x^2\).
Combine this with the LCM of the numbers 1 and 4, which is 4. Therefore, the LCM for \(x^2\) and \(4x\) becomes \(4x^2\). Knowing this will help you find a common ground where each fraction can meet.

Denominators

In rational expressions, the denominator is at the heart of fraction manipulation. It's important to manage and convert denominators to attain uniform denominators across multiple expressions.

• Your goal here is to transform each fraction so that all have the same denominator.
• This enables easy addition or subtraction of fractions later if needed.

For instance, when given the fractions \(\frac{9}{x^2}\) and \(\frac{1}{4x}\), your main focus is to convert their denominators to the LCM, which is \(4x^2\).
This process implies multiplying each term by whatever it needs to reflect this shared denominator. It's like finding a common language for them to communicate.

Algebraic Fractions

Algebraic fractions are like numerical fractions but involve variables along with numbers in their expressions. Understanding them will help you manipulate various algebraic terms as fractions.

• These fractions operate under the same basic fraction rules but with attention to variables as well.
• Often, operations on these fractions require factoring, finding LCM, or common denominators.

For example, to handle \(\frac{9}{x^2}\) and \(\frac{1}{4x}\), treat the variable parts as essential components of the expression.
Here, calculations must respect both numerical and algebraic aspects, which might involve factoring expressions or employing the distributive property in simplifications.

Fraction Conversion

Fraction conversion is the technique of changing the expression of fractions so they share a common denominator. This step is critical in simplifying or comparing fractions.

• The goal is to have a unified denominator for ease of operations, particularly addition or subtraction.
• This conversion involves multiplying both the numerator and the denominator of a fraction by the same number or expression.

In our example, the conversion requires \(\frac{9}{x^2}\) to become \(\frac{36}{4x^2}\) by multiplying it with \(\frac{4}{4}\), and \(\frac{1}{4x}\) to become \(\frac{x}{4x^2}\) by multiplying it with \(\frac{x}{x}\).
This adjustment helps the fractions to align under the same denominator, \(4x^2\), making future calculations straightforward.