Problem 6 - Elementary and Intermediate Algebra

Question

The LCD for \(\frac{1}{9 n^{2}}\) and \(\frac{37}{15 n^{3}}\) is \(3 \cdot 3 \cdot 5 \cdot n \cdot n \cdot n=45 n^{3}\) If we want to add these rational expressions, what form of 1 should be used a. to build \(\frac{1}{9 n^{2}} ?\) b. to build \(\frac{37}{15 n^{3}} ?\)

Step-by-Step Solution

Answer

Use \(\frac{5n}{5n}\) for \(\frac{1}{9n^2}\) and \(\frac{3}{3}\) for \(\frac{37}{15n^3}\).

Step 1: Understand the Problem

We need to find a form of 1 to multiply each fraction by so that both fractions have the same denominator, which is the LCD provided: \(45n^3\).

Step 2: Determine the Form of 1 for \(\frac{1}{9n^2}\)

The fraction \(\frac{1}{9n^2}\) needs the denominator \(45n^3\). We need to multiply the numerator and the denominator by the form of 1 that will change \(9n^2\) to \(45n^3\). Calculate the missing factors: \[\frac{45n^3}{9n^2} = \frac{5n}{1}\]. The form of 1 to use is \(\frac{5n}{5n}\).

Step 3: Determine the Form of 1 for \(\frac{37}{15n^3}\)

Already \(\frac{37}{15n^3}\) has the denominator \(15n^3\). We need to change this to \(45n^3\). Determine what to multiply to adjust the denominator: \[\frac{45n^3}{15n^3} = \frac{3}{1}\]. Therefore, use the form of 1 as \(\frac{3}{3}\).

Step 4: Answer the Question

We have found that the required form of 1 for \(\frac{1}{9n^2}\) is \(\frac{5n}{5n}\), and for \(\frac{37}{15n^3}\) is \(\frac{3}{3}\). These ensure both fractions have the common denominator when added.

Key Concepts

Least Common Denominator (LCD)

When dealing with fractions, especially rational expressions, it's crucial to understand the concept of the Least Common Denominator (LCD). The LCD is the smallest common multiple of the denominators of two or more fractions. This is essential in ensuring that all fractions involved in a calculation share the same base before performing any addition or subtraction.

To find the LCD, identify the denominators of the fractions you are working with. For instance, if you have fractions with denominators such as \(9n^2\) and \(15n^3\), you need to determine the smallest expression they can both divide into without leaving a remainder.

• Factor each denominator into its prime factors. Here, \(9n^2 = 3^2 \times n^2\) and \(15n^3 = 3 \times 5 \times n^3\).
• The next step is to take the highest power of each common factor. In this case, you'll get \(3^2\), \(5\), and \(n^3\).
• Multiply these highest common factors together to achieve the LCD, resulting in \(45n^3\) for this particular problem.

Having a common denominator allows for straightforward addition or subtraction of fractions, making the process seamless and preventing errors.

Adding Fractions

Once you've obtained the Least Common Denominator (LCD) for a set of fractions, adding them becomes much simpler. The process involves transforming each fraction to have this common denominator so they can be combined easily.

Here's how to add fractions using the LCD:

• Adjust each fraction by multiplying both the numerator and the denominator by a value that converts the denominator to the LCD while keeping the fraction's value unchanged.
• For example, changing \(\frac{1}{9n^2}\) to an equivalent form with the LCD \(45n^3\), involves multiplying by the factor \(\frac{5n}{5n}\).
• Similarly, for \(\frac{37}{15n^3}\), multiply by \(\frac{3}{3}\) to reach the common denominator \(45n^3\).

Once each fraction is expressed with the LCD as its denominator, adding them is simply a matter of adding their numerators. This uniformity ensures the result is accurate and consistent.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or portion of a whole. Understanding this concept is key when manipulating fractions to share a common denominator.

To create equivalent fractions, multiply the numerator and the denominator of a fraction by the same non-zero number. This keeps the ratio of the numerator to the denominator unchanged.

• Take \(\frac{1}{9n^2}\) as an example. To make it equivalent to a fraction with the LCD \(45n^3\), multiply by \(\frac{5n}{5n}\).
• This results in a new fraction \(\frac{5n}{45n^3}\), which is equivalent to the original fraction.
• Similarly, convert \(\frac{37}{15n^3}\) into an equivalent form with the LCD by multiplying by \(\frac{3}{3}\). The resulting \(\frac{111}{45n^3}\) maintains the original value of the fraction.

By understanding and applying the principle of creating equivalent fractions, you can easily transition from individual distinct fractions to those sharing a common denominator, which is essential for tasks such as adding or subtracting fractions.