Problem 2 - Elementary Algebra

Question

For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{3}{8} ; \frac{5}{6} $$

Step-by-Step Solution

Answer

The least common denominator is 24.

Step 1 - Prime Factorization of Denominators

Find the prime factors of the denominators 8 and 6. 8: \(8 = 2^3\) 6: \(6 = 2 \times 3\)

Step 2 - Identify the Highest Powers of Prime Factors

Determine the highest powers of all prime factors. For 8 and 6, the prime factors are 2 and 3. 2: highest power is \(2^3\) 3: highest power is \(3^1\)

Step 3 - Compute the Least Common Denominator (LCD)

Multiply the highest powers of all prime factors to find the LCD. LCD = \(2^3 \times 3^1 = 8 \times 3 = 24\)

Key Concepts

Prime Factorization

Prime factorization is a way of expressing a number as the product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example: 2, 3, 5, 7, 11, etc. By breaking down larger numbers into their prime factors, we can work with their basic building blocks.

Let’s dive into the concept with an example: Given the denominators 8 and 6, we convert them to their prime factors. For 8, we have:
\(8 = 2^3\). Here, 2 is prime and appears three times in multiplication. For 6, we have:
\(6 = 2 \times 3\). Both 2 and 3 are primes. Understanding this breakdown helps us simplify or compare fractions by working with their most fundamental components.

Highest Powers of Prime Factors

Once we have the prime factors, we need to determine the highest powers of these factors. This step is crucial to finding the least common denominator (LCD). The LCD is the smallest number that each denominator can divide evenly.

For the numbers we considered before, 8 and 6, the prime factors are 2 and 3. Look at the highest power of each prime factor appearing in the factorization:

• For 2, the highest power is \(2^3\)
• For 3, the highest power is \(3^1\)
Here, 2^3 (which is 8) is taken because 2 appears to the power of 3 in 8. For 3, we take 3^1 (which is 3) because 3 appears to the power of 1 in 6. By choosing the highest powers, we ensure that our common denominator can accommodate all the factors from the original denominators.

Multiplication of Prime Factors

After identifying the highest powers of each prime factor, the next step is to multiply them together. This step helps us find the least common denominator (LCD).

In our example, the highest powers are 2^3 and 3^1. To find the LCD, we multiply these together:
\br> \(LCD = 2^3 \times 3^1 \)

\br> Doing the multiplication, we get:
\( LCD = 8 \times 3 = 24 \) . Hence, the least common denominator of fractions with denominators 8 and 6 is 24. By breaking down the process into steps, prime factorization makes it easier to understand and compute the LCD, making fractions simpler to handle.