Problem 3 - Algebra 1

Question

Find the least common denominator. $$\frac{3}{4 x}, \frac{1}{6 x^{2}}, \frac{1}{8 x^{2}}$$

Step-by-Step Solution

Answer

The least common denominator for \(\frac{3}{4 x}, \frac{1}{6 x^{2}}, \frac{1}{8 x^{2}}\) is \(24x^2\).

Step 1: List Down the Factors

Start by listing down the prime factors of each denominator. \[4x = 2^2 \cdot x\], \[6x^2 = 2 \cdot 3 \cdot x^2\], \[8x^2 = 2^3\cdot x^2\].

Step 2: Identify Highest Powers

The next step is to identify the highest power of each prime number across the denominators. This is given as \[2^3, 3^1, x^2\]

Step 3: Calculate the Least Common Denominator

Finally, multiply these highest powers together to obtain the least common denominator: \[2^3 \cdot 3^1 \cdot x^2 = 24x^2\]

Key Concepts

Prime Factors

To find the least common denominator (LCD) of rational expressions like \( \frac{3}{4x} \), \( \frac{1}{6x^{2}} \), and \( \frac{1}{8x^{2}} \), we start by understanding the concept of prime factors. When breaking down numbers or expressions, prime factorization involves expressing a number as a product of prime numbers. A prime number is one that can only be divided by 1 and itself.

For example, in the case of the denominator \( 4x \), it can be expressed as the product of its prime factors: \( 4x = 2^2 \times x \) since 4 is \( 2^2 \) and \( x \) is already a prime factor in terms of variables.

This method is repeated for other denominators:

• For \( 6x^2 \), the prime factorization is \( 2 \times 3 \times x^2 \), as 6 is \( 2 \times 3 \) and \( x^2 \) reflects that \( x \) is a variable factor appearing twice.
• Similarly, \( 8x^2 \) can be expressed as \( 2^3 \times x^2 \), with 8 being \( 2^3 \).

By breaking each part of the expression into prime factors, you lay the groundwork to then find their greatest power, which is essential for determining the LCD.

Greatest Power

Once the prime factors are identified, the next step is finding the greatest power of each prime number present across all the denominators. This step ensures that each denominator can be expressed as a factor of the LCD, allowing us to compare and combine the rational expressions on the same terms.

Consider the prime factors identified earlier:

• \( \text{For the prime } 2, \text{the highest power across } 4x, 6x^2, \text{ and } 8x^2 \text{ is } 2^3. \)
• \( \text{For the prime } 3, \text{ it only appears in } 6x^2 \text{ so the highest power is } 3^1. \)
• \( \text{For the variable } x, \text{highest power is found in } 6x^2 \text{ and } 8x^2 \text{ which is } x^2. \)

By selecting the highest powers of each prime number and variable, we can build the least common denominator. In this example, as derived, the greatest powers yield \( 2^3 \), \( 3^1 \), and \( x^2 \).

These collectively ensure that we account for all necessary factors, without overcompensation, to establish the smallest possible common denominator needed for our rational expressions.

Rational Expressions

Rational expressions involve fractions that include polynomials in their numerators, denominators, or both. Handling rational expressions often requires manipulating these fractions to have a common denominator for operations such as addition or subtraction.

For the expressions \( \frac{3}{4x}, \frac{1}{6x^2}, \text{ and } \frac{1}{8x^2} \), finding the least common denominator facilitates unification for further arithmetic operations.

Here's the process simplified:

• First, decompose each denominator into its prime factors, as discussed previously.
• Then, identify the greatest power of each factor among the denominators.
• Finally, multiply these greatest powers to form the least common denominator.

In this example, we've determined the least common denominator to be \( 24x^2 \). This allows each rational expression to be rewritten over a common denominator, setting the stage for addition or subtraction of these expressions.

Understanding this process allows you to manage and manipulate rational expressions more effectively in any mathematical context involving fractions.